# Spectral representations of characteristic functions of discrete probability
laws

I. A. Alexeev111The Euler International Mathematical Institute, 10 Pesochnaya
nab., 197022 St. Petersburg, Russia, e-mail<EMAIL_ADDRESS>A. A.
Khartov222Smolensk State University, 4 Przhevalsky st., 214000 Smolensk,
Russia, e-mail<EMAIL_ADDRESS>
###### Abstract

We consider discrete probability laws on the real line, whose characteristic
functions are separated from zero. This class includes arbitrary discrete
infinitely divisible laws and lattice probability laws, whose characteristic
functions have no zeroes on the real line. We show that characteristic
functions of such laws admit spectral Lévy–Khinchine type representation with
non-monotonic Lévy spectral function. We also apply the representations of
such laws to obtain limit and compactness theorems with convergence in
variation to probability laws from this class.

Keywords and phrases: discrete probability laws, characteristic functions,
spectral Lévy-Khinchine type representations, quasi-infinitely divisible laws,
convergence in variation, relative compactness, stochastic compactness.

##  1 Introduction

This paper is devoted to the Lévy–Khinchine type representations for
characteristic functions of univariate discrete probability laws from rather
wide class.

Let $F$ be a distribution function of a probability law on the real line
$\mathbb{R}$. Recall that $F$ is called infinitely divisible if for every
positive integer $n$ there exists a distribution function $F_{1/n}$ such that
$F=(F_{1/n})^{*n}$, where “$*$” is the convolution, i.e. $F$ is $n$-fold
convolution power of $F_{1/n}$. It is well known that $F$ is infinitely
divisible if and only if its characteristic function

$\displaystyle f(t)\colonequals\int\limits_{\mathbb{R}}e^{itx}dF(x),\quad
t\in\mathbb{R},$

admits the following representation

$\displaystyle
f(t)=\exp\Biggr{\\{}it\gamma_{\tau}-\frac{\sigma^{2}t^{2}}{2}+\int\limits_{\mathbb{R}\setminus\\{0\\}}\bigl{(}e^{itu}-1-\tfrac{it}{\tau}\sin(\tau
u)\bigr{)}d\Lambda(u)\Biggr{\\}},\quad t\in\mathbb{R},$ (1)

for any $\tau>0$ and some $\gamma_{\tau}\in\mathbb{R}$, $\sigma\geqslant 0$,
and function $\Lambda$ (the Lévy spectral function), which is non-decreasing
on every interval $(-\infty,0)$ and $(0,+\infty)$, and it satisfies

$\displaystyle\lim_{u\to-\infty}\Lambda(u)=\lim_{u\to+\infty}\Lambda(u)=0,$
(2)

and also

$\displaystyle\int\limits_{0<|u|<\delta}x^{2}d\Lambda(u)<+\infty,\quad\delta>0.$

Here $\gamma_{\tau}$ depend on $\tau$, but $\sigma$ and $\Lambda$ are not. We
use $u\mapsto\tfrac{1}{\tau}\sin(\tau u)$ as the “centering function” in the
integral in (1) following to Zolotarev [26] and [27]. This function can be
replaced by any bounded Borel function $c:\mathbb{R}\to\mathbb{R}$, which
satysfies $c(0)=0$ and $(c(x)-x)/x^{2}\to 0$ as $x\to 0$ (see [17]).

It is well known that the properties of infinitely divisible $F$ (continuity,
moments, and support) can be described in terms of its triplet
$(\gamma_{\tau},\sigma^{2},\Lambda)$ (see [24]).

It turns out that there are probability laws, whose characteristic functions
admit representation of type (1) but with non-monotonic spectral function
$\Lambda$. The examples of such laws can be found in the classical monographs
[10], [18], and [19]. The corresponding class was introduced by Lindner and
Sato [16]. Following them, a distribution function $F$ is called quasi-
infinitely divisible if its characteristic function $f$ admits the
representation (1) with some (any) $\tau>0$ and some
$\gamma_{\tau}\in\mathbb{R}$, $\sigma\geqslant 0$, and function $\Lambda$,
which has finite total variation on every interval $(-\infty,-r]$ and
$[r,\infty)$, $r>0$, and it satisfies (2) and

$\displaystyle\int\limits_{0<|u|<\delta}x^{2}d|\Lambda|(u)<+\infty,\quad\delta>0.$

Such probability laws find interesting applications in theory of stochastic
processes (see [16] and [21]), number theory (see [20]), physics (see [6] and
[7]), insurance mathematics (see [25]).

The first detailed analysis of quasi-infinitely divisible distribution
functions on $\mathbb{R}$ was performed in [17] (the multivariate case is
considered in the recent papers [1] and [2]). In these works the authors
studied questions concerning supports, moments, continuity, and weak
convergence. The most complete results were obtained for probability laws on
the set of integers $\mathbb{Z}$ (with extension for lattice laws). The
following important fact was stated in [17].

###### Theorem 1

Let $F$ be the distribution function of probability law on $\mathbb{Z}$ with
characteristic function $f$. Then $F$ is quasi-infinitely divisible if and
only if $f$ does not have zeroes on the real line. In that case, $f$ admits
the representation

$\displaystyle
f(t)=\exp\biggl{\\{}it\gamma+\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\lambda_{k}(e^{itk}-1)\biggr{\\}},\quad
t\in\mathbb{R},$ (3)

where $\gamma\in\mathbb{Z}$, $\lambda_{k}\in\mathbb{R}$,
$k\in\mathbb{Z}\setminus\\{0\\}$, and
$\sum_{k\in\mathbb{Z}\setminus\\{0\\}}|\lambda_{k}|<\infty$.

Here we omitted some detailes from the original formulation of this theorem.
It is not difficult to check that (3) can be written in the form (1) (we will
do the similar derivation in the next section).

In [17] the authors also obtained a criterion of weak convergence for
sequences of quasi-infinitely divisible laws to a fixed law from this class in
terms of parameters of (3). These results were complemented by compactness
theorems in [11]. It is known that weak and total variation convergences are
equivalent for probability laws on $\mathbb{Z}$. Therefore these results can
be considered within the convergence in variation.

In this paper we generalize the above mentioned results. Namely, we state
similar facts for rather wide class of discrete probability laws on
$\mathbb{R}$, whose characteristic functions are separated from zero. In
particular, this class includes arbitrary discrete infinitely divisible laws
and lattice probability laws, whose characteristic functions have no zeroes on
the real line. The first step in this direction was made in [12].

##  2 Representation theorem

Let us consider an arbitrary discrete distribution function

$\displaystyle F(x)\colonequals\sum_{k=1}^{\infty}p_{x_{k}}I_{x_{k}}(x),\quad
x\in\mathbb{R},$

where $x_{k}$, $k\in\mathbb{N}$ (the set of positive integers), are distinct
real numbers, $p_{x_{k}}\geqslant 0$, $k\in\mathbb{N}$, and
$\sum_{k=1}^{\infty}p_{x_{k}}=1$. Here and below $I_{c}(x)=1$ if $x\geqslant
c$ and $I_{c}(x)=0$ if $x<c$ for any fixed $c\in\mathbb{R}$. Let $X$ be the
set of all points of growth of $F$, i.e.
$X\colonequals\\{x_{k}:p_{x_{k}}>0,\,k\in\mathbb{N}\\}\neq\varnothing$. So we
can write

$\displaystyle F(x)=\sum_{x_{k}\in X}p_{x_{k}}I_{x_{k}}(x),\quad
x\in\mathbb{R}.$

Thus $F$ corresponds to a discrete probability law concentrated on the
countable set $X$ of real numbers.

Let $f$ be the characteristic function of $F$. Then

$\displaystyle f(t)=\sum_{k=1}^{\infty}p_{x_{k}}e^{itx_{k}}=\sum_{x_{k}\in
X}p_{x_{k}}e^{itx_{k}},\quad t\in\mathbb{R}.$

So $f$ is an almost periodic function with absolutely convergent Fourier
series.

We focus on the case, when $f$ is separated from zero. Namely, we always
assume that

$\text{{there exists}\quad}\mu>0\quad\text{{such
that}}\quad|f(t)|\geqslant\mu>0\text{\quad{for all}\quad}t\in\mathbb{R}.$
$None$

Condition $(S)$ is equivalent to the relation $\inf_{t\in\mathbb{R}}|f(t)|>0$.

Let $\boldsymbol{D}$ denote the class of all discrete distribution functions.
Let $\boldsymbol{D}_{\\!S}$ be the subclass of $\boldsymbol{D}$, which
consists of distribution functions, whose characteristic functions satisfy
condition $(S)$. The class $\boldsymbol{D}_{\\!S}$ is rather wide. For
example, if there exists $x_{k_{0}}$ such that $p_{x_{k_{0}}}>1/2$, then
$F\in\boldsymbol{D}_{\\!S}$. This is seen from the following estimate:

$\displaystyle|f(t)|$ $\displaystyle=$
$\displaystyle\biggl{|}p_{x_{k_{0}}}e^{itx_{k_{0}}}+\sum_{k\in\mathbb{N}:\,k\neq
k_{0}}p_{x_{k}}e^{itx_{k}}\biggr{|}\geqslant
p_{x_{k_{0}}}-\sum_{k\in\mathbb{N}:\,k\neq k_{0}}p_{x_{k}}$
$\displaystyle{}\qquad=p_{x_{k_{0}}}-(1-p_{x_{k_{0}}})=2p_{x_{k_{0}}}-1>0.$

Moreover, $\boldsymbol{D}_{\\!S}$ contains all distribution functions of
lattice probability laws, whose characteristic functions have no zeroes on the
real line, i.e. if $X\subset\\{a+bl:l\in\mathbb{Z}\\}$ with some
$a\in\mathbb{R}$ and $b>0$ and also $f(t)\neq 0$, $t\in\mathbb{R}$, then
$F\in\boldsymbol{D}_{\\!S}$. Indeed, in this case $|f(\cdot)|$ is a continuous
$2\pi/b$-periodic function and its infimum is attained at some point in the
period segment. Thus to have no zeroes on the real line means that
$\inf_{t\in\mathbb{R}}|f(t)|>0$.

It is important to note that all discrete infinitely divisible distribution
functions belongs to $\boldsymbol{D}_{\\!S}$. Let us show this. It is known
(see [4]) that discrete distribution function $F$ is infinitely divisible if
and only if its characteristic function $f$ admits the following
representation

$\displaystyle
f(t)=\exp\biggr{\\{}it\gamma+\sum_{k=1}^{\infty}\lambda_{u_{k}}(e^{itu_{k}}-1)\biggr{\\}},\quad
t\in\mathbb{R},$ (4)

where $\gamma\in\mathbb{R}$, $u_{k}\in\mathbb{R}$ and
$\lambda_{u_{k}}\geqslant 0$ for all $k\in\mathbb{N}$, and
$\sum_{k=1}^{\infty}\lambda_{u_{k}}<\infty$. Therefore we have the estimate

$\displaystyle|f(t)|=\exp\biggr{\\{}\sum_{k=1}^{\infty}\lambda_{u_{k}}(\cos(tu_{k})-1)\biggr{\\}}\geqslant\exp\biggl{\\{}-2\sum_{k=1}^{\infty}\lambda_{u_{k}}\biggr{\\}}>0,\quad
t\in\mathbb{R}.$

We see that $f$ satisfies $(S)$, i.e. $F\in\boldsymbol{D}_{\\!S}$.

There exist examples of characteristic functions of probability laws admitting
representation (4), where some of $\lambda_{u_{k}}$ can be negative within
condition $\sum_{k=1}^{\infty}|\lambda_{u_{k}}|<\infty$ (see [10] pp. 81–83,
[18] p. 165, and [19] p. 123–124). For such characteristic functions we
similarly get the estimate

$\displaystyle|f(t)|\geqslant\exp\biggl{\\{}-2\sum_{k=1}^{\infty}|\lambda_{u_{k}}|\biggr{\\}}>0,\quad
t\in\mathbb{R}.$

Thus the corresponding distribution functions belong to
$\boldsymbol{D}_{\\!S}$. In the result below we show that the converse fact is
also true, i.e. the characteristic function of every
$F\in\boldsymbol{D}_{\\!S}$ admits representation of type (4) with
$\gamma\in\mathbb{R}$, $\lambda_{u_{k}}\in\mathbb{R}$, $k\in\mathbb{N}$, and
$\sum_{k=1}^{\infty}|\lambda_{u_{k}}|<\infty$.

Howeever, it is clear that $\boldsymbol{D}_{\\!S}\neq\boldsymbol{D}$. For
example, the distribution function
$G(x)\colonequals\tfrac{1}{2}I_{0}(x)+\tfrac{1}{2}I_{1}(x)$, $x\in\mathbb{R}$,
does not belong to $\boldsymbol{D}_{\\!S}$. Indeed, its characteristic
function $g(t)=\tfrac{1}{2}+\tfrac{1}{2}e^{it}$, $t\in\mathbb{R}$, has zeroes
in the points $\pi(2k+1)$, $k\in\mathbb{Z}$. There are exist more interesting
examples. Namely, let us consider the distribution function
$H(x)\colonequals\tfrac{1}{2}I_{0}+p_{\alpha}I_{\alpha}(x)+p_{1}I_{1}(x)$,
$x\in\mathbb{R}$, where $\alpha$ is an irrational number from $(0,1)$, the
weights $p_{\alpha}$ and $p_{1}$ are strictly positive, and
$p_{\alpha}+p_{1}=1/2$. Its characteristic function
$h(t)=\tfrac{1}{2}+p_{\alpha}e^{it\alpha}+p_{1}e^{it}$, $t\in\mathbb{R}$, does
not have zeroes, and it is not separated from zero, i.e. $h(t)\neq 0$,
$t\in\mathbb{R}$, and $\inf_{t\in\mathbb{R}}|h(t)|=0$. Thus we have
$H\notin\boldsymbol{D}_{\\!S}$. This and other similar particular results were
obtained in [12].

For sharp formulation of the result below we need to introduce the set of all
finite $\mathbb{Z}$–linear combinations of elements from a set
$Y\subset\mathbb{R}$:

$\displaystyle\langle
Y\rangle\colonequals\biggl{\\{}\sum_{k=1}^{m}c_{k}y_{k}:\,c_{k}\in\mathbb{Z},\,y_{k}\in
Y,\,m\in\mathbb{N}\biggr{\\}}.$

In other words, $\langle Y\rangle$ is a module over the ring $\mathbb{Z}$ with
the generating set $Y$. It easily seen that, in particular, $Y\subset\langle
Y\rangle$, $0\in\langle Y\rangle$, and $ky\in\langle Y\rangle$ for any
$k\in\mathbb{Z}$, $y\in Y$. If $Y\neq\\{0\\}$, then $\langle Y\rangle$ is an
infinite countable set.

We always set below that sums over the null set equal zero.

###### Theorem 2

Under the assumption $(S)$, the characteristic function $f$ admits the
following representation

$\displaystyle f(t)=\exp\biggr{\\{}it\gamma+\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}(e^{itu}-1)\biggr{\\}},\quad
t\in\mathbb{R},$ (5)

where $\gamma\in\langle X\rangle$, $\lambda_{u}\in\mathbb{R}$ for all
$u\in\langle X\rangle\setminus\\{0\\}$, and $\sum_{u\in\langle
X\rangle\setminus\\{0\\}}|\lambda_{u}|<\infty$.

Before the proof of Theorem 2 we will do some remarks.

According to this theorem, if $F\in\boldsymbol{D}_{\\!S}$, i.e. $f$ satisfies
$(S)$, then $f$ admits representation (5). Let $\mathrm{Arg}\,f(t)$ be the
argument of $f(t)$ (i.e. $f(t)=\exp\\{i\mathrm{Arg}\,f(t)\\}$) that is
uniquely defined for $t\in\mathbb{R}$ by continuity with the condition
$\mathrm{Arg}\,f(0)=0$. Then

$\displaystyle\mathrm{Arg}\,f(t)=t\gamma+\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}\sin(tu),\quad t\in\mathbb{R}.$

Due to $\sum_{u\in\langle X\rangle\setminus\\{0\\}}|\lambda_{u}|<\infty$, we
have

$\displaystyle\gamma=\lim\limits_{T\to\infty}\dfrac{\mathrm{Arg}\,f(T)}{T}=\lim\limits_{T\to\infty}\dfrac{\mathrm{Arg}\,f(T)-\mathrm{Arg}\,f(0)}{T-0}.$
(6)

Thus $\gamma$ is a “mean motion” of the argument of $f$ on $[0,\infty)$.

Let us fix arbitrarily $\tau>0$ and introduce “mean motion” on the interval
$[0,\tau]$:

$\displaystyle\gamma_{\tau}\colonequals\dfrac{\mathrm{Arg}\,f(\tau)}{\tau}=\gamma+\tfrac{1}{\tau}\\!\\!\\!\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}\sin(\tau u).$ (7)

Then we write

$\displaystyle f(t)=\exp\biggl{\\{}it\gamma_{\tau}+\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}(e^{itu}-1-i\tfrac{t}{\tau}\sin(\tau
u))\biggr{\\}},\quad t\in\mathbb{R}.$

Let us define

$\displaystyle\Lambda(u)=\sum\limits_{k\in\mathbb{N}:u_{k}\leqslant
u}\lambda_{u_{k}},\quad
u<0,\quad\text{and}\quad\Lambda(u)=-\sum\limits_{k\in\mathbb{N}:u_{k}>u}\lambda_{u_{k}},\quad
u>0.$

Hence we have

$\displaystyle
f(t)=\exp\Biggr{\\{}it\gamma_{\tau}+\int\limits_{\mathbb{R}\setminus\\{0\\}}\bigl{(}e^{itu}-1-\tfrac{it}{\tau}\sin(\tau
u)\bigr{)}d\Lambda(u)\Biggr{\\}},\quad t\in\mathbb{R}.$

So we proved the following fact.

###### Remark 1

If $F\in\boldsymbol{D}_{\\!S}$, then it is quasi-infinitely divisible with the
triplet $(\gamma_{\tau},0,\Lambda)$.

It is easily seen that the function $\Lambda$ satisfies all the required
conditions.

Note that the centering by $u\mapsto\tfrac{1}{\tau}\sin(\tau u)$ in the
integral has the advantage that the shift parameter $\gamma_{\tau}$ of the
triplet gets additional meaning according to formulas (6) and (7).

For fixed $\tau>0$ the parameter $\gamma_{\tau}$ and the spectral function
$\Lambda$ are uniquely determined from $f$ (see [10], p. 80). Therefore
$\gamma$ and the sequence $\lambda_{u}$, $u\in\langle
X\rangle\setminus\\{0\\}$, are also uniquely determined in (5).

According to Remark 1 and comments above concerning lattice probability laws,
Theorem 2 is a generalization of Theorem 1.

It is very interesting to know whether or not every discrete quasi-infinitely
divisible distribution function belongs to $\boldsymbol{D}_{\\!S}$. It is
known that characteristic functions of quasi-infinitely divisible distribution
functions have no zeroes on the real line (see [17]). So the problem is
reduced to the following question. Are there exist discrete quasi-infinitely
divisible distribution functions with characteristic functions that have no
zeroes on the real line and that are not separated from zero? There are only
particular results. For example, the above mentioned distribution function $H$
is not quasi-infinitely divisible that was shown in [12].

Proof of Theorem 2. The proof is devided into two steps: for the case of
finite $X$ and for the general case.

1\. Suppose that $X$ is finite. Without loss of generality we can assume that
$X=\\{x_{1},\ldots,x_{n}\\}$. Hence $f(t)=\sum_{k=1}^{n}p_{x_{k}}e^{itx_{k}}$,
$t\in\mathbb{R}$. The case $n=1$ and $x_{1}=0$ is trivial: $\langle
X\rangle=\\{0\\}$, $\gamma=0$. We next exclude this case. Let
$\alpha_{1},\ldots,\alpha_{d}\in X$ be a basis in $X\neq\\{0\\}$ over
$\mathbb{Z}$, i.e. for every $k=1,\ldots,n$ there is the unique representation
$x_{k}=\sum_{j=1}^{d}c_{k,j}\alpha_{j}$ with $c_{k,j}\in\mathbb{Z}$. Here the
numbers $\alpha_{1},\ldots,\alpha_{d}$ are linearly independent over
$\mathbb{Z}$, that is the equation $l_{1}\alpha_{1}+\ldots+l_{1}\alpha_{d}=0$
with $l_{1},\ldots,l_{d}\in\mathbb{Z}$ holds if and only if
$l_{1}=l_{2}=\ldots=l_{d}=0$. It is easy to check that

$\displaystyle\langle
X\rangle=\biggl{\\{}\sum_{j=1}^{d}l_{j}\alpha_{j}:\,l_{j}\in\mathbb{Z}\biggr{\\}}.$
(8)

Using the representations for $x_{k}$, we consider the function

$\displaystyle\varphi(t_{1},\ldots,t_{d})=\sum_{k=1}^{n}p_{x_{k}}\exp\biggl{\\{}i\sum\limits_{j=1}^{d}c_{k,j}\alpha_{j}t_{j}\biggr{\\}},\quad
t_{1},\ldots,t_{d}\in\mathbb{R}.$

The function $\varphi$ is a continuous periodic function with periods
$2\pi/\alpha_{1},\ldots,2\pi/\alpha_{d}$ over variables $t_{1},\ldots,t_{d}$,
correspondingly. The function $f$ is the diagonal function for $\varphi$:

$\displaystyle\varphi(t,\ldots,t)=\sum_{k=1}^{n}p_{x_{k}}\exp\biggl{\\{}it\sum\limits_{j=1}^{d}c_{k,j}\alpha_{j}\biggr{\\}}=\sum_{k=1}^{n}p_{x_{k}}e^{itx_{k}}=f(t),\quad
t\in\mathbb{R}.$

Therefore the set of values $\\{f(t):t\in\mathbb{R}\\}$ is everywhere dense in
the set $\\{\varphi(t_{1},\ldots,t_{d}):t_{1},\ldots,t_{d}\in\mathbb{R}\\}$
(see [15], Theorem 5, p. 46). Hence, by the assumption $(S)$, we have

$\displaystyle|\varphi(t_{1},\ldots,t_{d})|\geqslant\mu>0,\quad
t_{1},\ldots,t_{d}\in\mathbb{R}.$ (9)

We now consider the following function

$\displaystyle\tilde{\varphi}(t_{1},\ldots,t_{d})\colonequals\varphi(t_{1}/\alpha_{1},\ldots,t_{d}/\alpha_{d})=\sum_{k=1}^{n}p_{x_{k}}\exp\biggl{\\{}i\sum\limits_{j=1}^{d}c_{k,j}t_{j}\biggr{\\}},\quad
t_{1},\ldots,t_{d}\in\mathbb{R}.$

It is the characteristic function for the probabilty law on the vectors
$(c_{k,1},\ldots,c_{k,d})\in\mathbb{Z}^{d}$ with weights $p_{x_{k}}$,
$k=1,\ldots,n$. Due to (9), we have
$|\tilde{\varphi}(t_{1},\ldots,t_{d})|\geqslant\mu>0$,
$t_{1},\ldots,t_{d}\in\mathbb{R}$. By Theorem 3.2 in [2], the function
$\tilde{\varphi}$ admits the following representation for any
$t_{1},\ldots,t_{d}\in\mathbb{R}$:

$\displaystyle\tilde{\varphi}(t_{1},\ldots,t_{d})$ $\displaystyle=$
$\displaystyle\exp\biggl{\\{}i(t_{1}\gamma_{1}+\ldots+t_{d}\gamma_{d})$
$\displaystyle{}\qquad+\\!\\!\\!\sum_{(l_{1},\ldots,l_{d})\in\mathbb{Z}^{d}\setminus\\{0\\}}\\!\\!\\!\lambda_{l_{1},\ldots,l_{d}}\Bigr{(}\exp\bigl{\\{}i(t_{1}l_{1}+\ldots+t_{d}l_{d})\bigr{\\}}-1\Bigr{)}\biggr{\\}},$

where $\gamma_{1},\ldots,\gamma_{d}\in\mathbb{Z}$,
$\lambda_{l_{1},\ldots,l_{d}}\in\mathbb{R}$ for
$(l_{1},\ldots,l_{d})\in\mathbb{Z}^{d}\setminus\\{0\\}$, and
$\sum_{(l_{1},\ldots,l_{d})\in\mathbb{Z}^{d}\setminus\\{0\\}}|\lambda_{l_{1},\ldots,l_{d}}|<\infty$.

Next, observe that

$\displaystyle\tilde{\varphi}(t\alpha_{1},\ldots,t\alpha_{d})=\varphi(t,\ldots,t)=f(t),\quad
t\in\mathbb{R}.$

Then for $t\in\mathbb{R}$ we have

$\displaystyle
f(t)=\exp\biggl{\\{}it(\gamma_{1}\alpha_{1}+\ldots+\gamma_{d}\alpha_{d})+\\!\\!\\!\sum_{(l_{1},\ldots,l_{d})\in\mathbb{Z}^{d}\setminus\\{0\\}}\\!\\!\\!\lambda_{l_{1},\ldots,l_{d}}\Bigr{(}\exp\bigl{\\{}it(l_{1}\alpha_{1}+\ldots+l_{d}\alpha_{d})\bigr{\\}}-1\Bigr{)}\biggr{\\}}.$

Due to (8), this representation we can write in the form (5), where
$\gamma\colonequals\gamma_{1}\alpha_{1}+\ldots+\gamma_{d}\alpha_{d}\in\langle
X\rangle$, and $\lambda_{u}\colonequals\lambda_{l_{1},\ldots,l_{d}}$ for
$u=l_{1}\alpha_{1}+\ldots+l_{d}\alpha_{d}\in\langle X\rangle\setminus\\{0\\}$
with $(l_{1},\ldots,l_{d})\in\mathbb{Z}^{d}\setminus\\{0\\}$.

2\. We now turn to the general case for $X$. So
$f(t)=\sum_{k=1}^{\infty}p_{x_{k}}e^{itx_{k}}$, $t\in\mathbb{R}$. Since
$X\neq\varnothing$, without loss of generality we can suppose that $x_{1}\in
X$, i.e. $p_{x_{1}}>0$. We will uniformly approximate $f$ by the following
characteristic functions:

$\displaystyle f_{n}(t)\colonequals\sum_{k=1}^{n}p_{n,x_{k}}e^{itx_{k}},\quad
t\in\mathbb{R},\quad n\in\mathbb{N},$

where

$\displaystyle
p_{n,x_{k}}\colonequals\dfrac{p_{x_{k}}}{\sum_{m=1}^{n}p_{x_{m}}},\quad
k=1,\ldots,n,\quad n\in\mathbb{N}.$

Here $\sum_{m=1}^{n}p_{x_{m}}\geqslant p_{x_{1}}>0$, $n\in\mathbb{N}$. Let us
estimate the approximation error for every $n\in\mathbb{N}$:

$\displaystyle\sup\limits_{t\in\mathbb{R}}|f(t)-f_{n}(t)|$ $\displaystyle=$
$\displaystyle\sup\limits_{t\in\mathbb{R}}\biggl{|}\sum_{k=1}^{n}(p_{x_{k}}-p_{n,x_{k}})e^{itx_{k}}+\sum_{k=n+1}^{\infty}p_{x_{k}}e^{itx_{k}}\biggr{|}$
$\displaystyle\leqslant$
$\displaystyle\sum_{k=1}^{n}|p_{x_{k}}-p_{n,x_{k}}|+\sum_{k=n+1}^{\infty}p_{x_{k}},$

where, due to the equality $\sum_{m=1}^{\infty}p_{x_{m}}=1$, we have

$\displaystyle\sum_{k=1}^{n}|p_{x_{k}}-p_{n,x_{k}}|=\sum_{k=1}^{n}p_{x_{k}}\biggl{(}\dfrac{1}{\sum_{m=1}^{n}p_{x_{m}}}-1\biggr{)}=\sum_{k=1}^{n}p_{x_{k}}\biggl{(}\dfrac{\sum_{m=n+1}^{\infty}p_{x_{m}}}{\sum_{m=1}^{n}p_{x_{m}}}\biggr{)}=\sum_{m=n+1}^{\infty}p_{x_{m}}.$

Thus we obtain

$\displaystyle\sup\limits_{t\in\mathbb{R}}|f(t)-f_{n}(t)|\leqslant
2\sum_{k=n+1}^{\infty}p_{x_{k}},\quad n\in\mathbb{N}.$

From this we conclude that

$\displaystyle\sup_{t\in\mathbb{R}}|f(t)-f_{n}(t)|\to 0,\quad n\to\infty.$

Let us take $\mu>0$ from $(S)$, and fix arbitrarily $\varepsilon\in(0,1/4)$.
Then there exists $n_{\varepsilon}\in\mathbb{N}$ such that

$\displaystyle\sup_{t\in\mathbb{R}}|f(t)-f_{n}(t)|\leqslant\varepsilon\mu,\quad
n\geqslant n_{\varepsilon}.$ (10)

From this we see that $f_{n}(t)\neq 0$, $t\in\mathbb{R}$, for every
$n\geqslant n_{\varepsilon}$. Indeed,

$\displaystyle|f_{n}(t)|\geqslant|f(t)|-|f(t)-f_{n}(t)|\geqslant\inf_{t\in\mathbb{R}}|f(t)|-\sup_{t\in\mathbb{R}}|f(t)-f_{n}(t)|,$

and, due to $(S)$ and (10), we have

$\displaystyle|f_{n}(t)|\geqslant(1-\varepsilon)\mu>0,\quad t\in\mathbb{R}.$
(11)

Choose $n\geqslant n_{\varepsilon}$ and represent $f(t)=f_{n}(t)\cdot
R_{n}(t)$ with $R_{n}(t)\colonequals f(t)/f_{n}(t)$, $t\in\mathbb{R}$. Since
$f$, $f_{n}$, $R_{n}$ are continuous functions without zeroes and they equal
$1$ at $t=0$, we can proceed to the distinguished logarithms:

$\displaystyle\mathrm{Ln}\,f(t)=\mathrm{Ln}\,f_{n}(t)+\mathrm{Ln}\,R_{n}(t),\quad
t\in\mathbb{R},$ (12)

which are uniquely defined by continuity with the conditions that they equal
$0$ at $t=0$ (see [13] p. 14–16 or [22] p. 2). We next consider
$\mathrm{Ln}\,f_{n}$ and $\mathrm{Ln}\,R_{n}$ separately.

By the part 1., for the function $f_{n}$ we have

$\displaystyle f_{n}(t)=\exp\biggl{\\{}it\gamma_{n}+\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}(e^{itu}-1)\biggr{\\}},\quad
t\in\mathbb{R},$

where $X_{n}=\bigl{\\{}x_{k}:p_{x_{k}}>0,\,k=1,\ldots,n\bigr{\\}}$,
$\gamma_{n}\in\langle X_{n}\rangle$, $\lambda_{n,u}\in\mathbb{R}$ for all
$u\in\langle X_{n}\rangle\setminus\\{0\\}$, and $\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}|\lambda_{n,u}|<\infty$. Since the function in
the exponent is continuous and it equals $0$ at $t=0$, we have

$\displaystyle\mathrm{Ln}\,f_{n}(t)=it\gamma_{n}+\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}(e^{itu}-1),\quad t\in\mathbb{R}.$

Setting $\lambda_{n,0}\colonequals-\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}$ , we represent $f_{n}$ in the
following form

$\displaystyle\mathrm{Ln}\,f_{n}(t)=it\gamma_{n}+\sum_{u\in\langle
X_{n}\rangle}\lambda_{n,u}e^{itu},\quad t\in\mathbb{R}.$

Observe that $X_{n}\subset X$, and hence $\langle X_{n}\rangle\subset\langle
X\rangle$. Then we can write

$\displaystyle\mathrm{Ln}\,f_{n}(t)=it\gamma_{n}+\sum_{u\in\langle
X\rangle}\lambda_{n,u}e^{itu},\quad t\in\mathbb{R},$ (13)

where for every $u\in\langle X\rangle\setminus\langle X_{n}\rangle$ we define
$\lambda_{n,u}\colonequals 0$ if $\langle X\rangle\setminus\langle
X_{n}\rangle\neq\varnothing$.

Next, we consider the function $\mathrm{Ln}\,R_{n}$. Observe that

$\displaystyle\mathrm{Ln}\,R_{n}(t)=\ln\biggl{(}1+\dfrac{f(t)-f_{n}(t)}{f_{n}(t)}\biggr{)},\quad
t\in\mathbb{R},$ (14)

(the latter is the principal value of the logarithm) because due to (10) and
(11)

$\displaystyle\sup_{t\in\mathbb{R}}\biggl{|}\dfrac{f(t)-f_{n}(t)}{f_{n}(t)}\biggr{|}\leqslant\dfrac{\sup_{t\in\mathbb{R}}|f(t)-f_{n}(t)|}{\inf_{t\in\mathbb{R}}|f_{n}(t)|}\leqslant\dfrac{\varepsilon}{1-\varepsilon}<1,$
(15)

and the function in the right-hand side of (14) is continuous and equals $0$
at $t=0$. Using this, we get the decomposition

$\displaystyle\mathrm{Ln}\,R_{n}(t)=\sum\limits_{m=1}^{\infty}\dfrac{(-1)^{m-1}}{m}\biggl{(}\dfrac{f(t)-f_{n}(t)}{f_{n}(t)}\biggr{)}^{m},\quad
t\in\mathbb{R}.$

It yields the estimate

$\displaystyle\sup_{t\in\mathbb{R}}|\mathrm{Ln}\,R_{n}(t)|\leqslant\sum\limits_{m=1}^{\infty}\dfrac{1}{m}\,\sup_{t\in\mathbb{R}}\biggl{|}\dfrac{f(t)-f_{n}(t)}{f_{n}(t)}\biggr{|}^{m}\leqslant\sum\limits_{m=1}^{\infty}\dfrac{1}{m}\biggl{(}\dfrac{\varepsilon}{1-\varepsilon}\biggr{)}^{m}.$

Since $\varepsilon\in(0,1/4)$, here we have

$\displaystyle\sum\limits_{m=1}^{\infty}\dfrac{1}{m}\biggl{(}\dfrac{\varepsilon}{1-\varepsilon}\biggr{)}^{m}\leqslant\sum\limits_{m=1}^{\infty}\biggl{(}\dfrac{\varepsilon}{1-\varepsilon}\biggr{)}^{m}=\dfrac{\tfrac{\varepsilon}{1-\varepsilon}}{1-\tfrac{\varepsilon}{1-\varepsilon}}=\dfrac{\varepsilon}{1-2\varepsilon}<2\varepsilon.$

Thus we obtain

$\displaystyle\sup_{t\in\mathbb{R}}|\mathrm{Ln}\,R_{n}(t)|<2\varepsilon.$ (16)

Let us consider the function $(f-f_{n})/f_{n}$ from (14). It is clear that
$f-f_{n}$ is an almost periodic function with absolutely convergent Fourier
series. Due to (11), the function $1/f_{n}$ is also almost periodic (see [5]
p. 35 or [14] p. 21), and it has absolutely convergent Fourier series (see [9]
p. 175). These facts can be also shown using (13). Therefore the function
$(f-f_{n})/f_{n}$ is almost periodic with absolutely convergent Fourier
series.

Let us return to (14). Since the function $z\mapsto\ln(1+z)$, $z\in\mathbb{C}$
(the set of complex numbers), is analytic on the unit disk, due to (15) and
the above remark, we get that $\mathrm{Ln}\,R_{n}$ is almost periodic with
absolute convergent Fourier series (see [9] p. 175):

$\displaystyle\mathrm{Ln}\,R_{n}(t)=\sum\limits_{u\in\Delta_{n}}\beta_{n,u}e^{itu},\quad
t\in\mathbb{R},$ (17)

where $\Delta_{n}$ is a countable set of real numbers (and, probably,
$\Delta_{n}\neq\langle X\rangle$), $\beta_{n,u}\in\mathbb{C}$ for
$u\in\Delta_{n}$, and $\sum_{u\in\Delta_{n}}|\beta_{n,u}|<\infty$. Without
loss of generality we can assume that either $\Delta_{n}=\varnothing$ (the
case $f=f_{n}$) and the sum equals zero in (17), or
$\Delta_{n}\neq\varnothing$ and $\beta_{n,u}\neq 0$, $u\in\Delta_{n}$.

We now return to the function $\mathrm{Ln}\,f$ and (12). The formulas (13) and
(17) yield

$\displaystyle\mathrm{Ln}\,f(t)=it\gamma_{n}+\sum_{u\in\langle
X\rangle}\lambda_{n,u}e^{itu}+\sum\limits_{u\in\Delta_{n}}\beta_{n,u}e^{itu},\quad
t\in\mathbb{R}.$

Recall that this is valid for every $n\geqslant n_{\varepsilon}$. Since
$\sum_{u\in\langle X\rangle}|\lambda_{n,u}|<\infty$ and
$\sum_{u\in\Delta_{n}}|\beta_{n,u}|<\infty$, $n\geqslant n_{\varepsilon}$, it
is true that

$\displaystyle\lim\limits_{T\to\infty}\dfrac{\mathrm{Ln}\,f(T)}{iT}=\gamma_{n},\quad
n\geqslant n_{\varepsilon}.$

Consequently, $\gamma_{n}$ are equal for $n\geqslant n_{\varepsilon}$, and we
denote their value by $\gamma$. Since $\gamma_{n}\in\langle
X_{n}\rangle\subset\langle X\rangle$, we have $\gamma\in\langle X\rangle$.

We next consider the function

$\displaystyle\mathrm{Ln}\,f(t)-it\gamma=\sum_{u\in\langle
X\rangle}\lambda_{n,u}e^{itu}+\sum\limits_{u\in\Delta_{n}}\beta_{n,u}e^{itu},\quad
t\in\mathbb{R},\quad n\geqslant n_{\varepsilon}.$ (18)

It is an almost periodic function with absolutely convergent Fourier series.
We define $\Delta=\cup_{n=n_{\varepsilon}}^{\infty}\Delta_{n}$, and we
introduce the Fourier coefficients

$\displaystyle\lambda_{v}\colonequals\lim\limits_{T\to\infty}\dfrac{1}{2T}\int\limits_{-T}^{T}\bigl{(}\mathrm{Ln}\,f(t)-it\gamma\bigr{)}e^{-itv}dt,\quad
v\in\langle X\rangle\cup\Delta.$ (19)

According to (17) and (18), for any $n\geqslant n_{\varepsilon}$ we have

$\displaystyle\lambda_{v}=\lim\limits_{T\to\infty}\dfrac{1}{2T}\int\limits_{-T}^{T}\biggl{(}\,\sum_{u\in\langle
X\rangle}\lambda_{n,u}e^{itu}\biggr{)}e^{-itv}dt+\lim\limits_{T\to\infty}\dfrac{1}{2T}\int\limits_{-T}^{T}\mathrm{Ln}\,R_{n}(t)e^{-itv}dt,\quad
v\in\langle X\rangle\cup\Delta.$

Due to (16), for any $n\geqslant n_{\varepsilon}$

$\displaystyle\Biggl{|}\dfrac{1}{2T}\int\limits_{-T}^{T}\mathrm{Ln}\,R_{n}(t)e^{-itv}dt\Biggr{|}\leqslant\dfrac{1}{2T}\int\limits_{-T}^{T}|\mathrm{Ln}\,R_{n}(t)|dt\leqslant\sup_{t\in\mathbb{R}}|\mathrm{Ln}\,R_{n}(t)|\leqslant
2\varepsilon.$

Since $\varepsilon>0$ can be choosen arbitrarily small, we obtain

$\displaystyle\lambda_{v}=\lim_{n\to\infty}\lim\limits_{T\to\infty}\dfrac{1}{2T}\int\limits_{-T}^{T}\biggl{(}\,\sum_{u\in\langle
X\rangle}\lambda_{n,u}e^{itu}\biggr{)}e^{-itv}dt,\quad v\in\langle
X\rangle\cup\Delta.$

The series in the integral is absolutely convergent. Hence

$\displaystyle\lambda_{v}=\lim_{n\to\infty}\sum_{u\in\langle
X\rangle}\lambda_{n,u}\biggl{(}\,\lim\limits_{T\to\infty}\dfrac{1}{2T}\int\limits_{-T}^{T}e^{itu}e^{-itv}dt\biggr{)},\quad
v\in\langle X\rangle\cup\Delta,$ (20)

where

$\displaystyle\lim\limits_{T\to\infty}\dfrac{1}{2T}\int\limits_{-T}^{T}e^{itu}e^{-itv}dt=\begin{cases}1,&u=v,\\\
0,&u\neq v.\end{cases}$ (21)

Suppose that $\Delta\setminus\langle X\rangle\neq\varnothing$, and
$v\in\Delta\setminus\langle X\rangle$. Then, on the one hand, from (20) and
(21) we have $\lambda_{v}=0$. On the other hand,
$v\in\Delta_{n}\setminus\langle X\rangle$ for some $n$. So from (18), (19),
and (21) we have $\lambda_{v}=\beta_{n,v}$ for such $n$, where
$\beta_{n,v}\neq 0$ according to the convention above. Thus we obtain a
contradiction. Hence $\Delta\subset\langle X\rangle$. Therefore

$\displaystyle\mathrm{Ln}\,f(t)-it\gamma=\sum_{u\in\langle
X\rangle}(\lambda_{n,u}+\beta_{n,u})e^{itu},\quad t\in\mathbb{R},\quad
n\geqslant n_{\varepsilon},$

where $\beta_{n,u}\colonequals 0$, $u\in\langle X\rangle\setminus\Delta_{n}$.
So $\lambda_{u}=\lambda_{n,u}+\beta_{n,u}$, $u\in\langle X\rangle$,
$n\geqslant n_{\varepsilon}$, and, from (20) and (21), we have
$\lambda_{u}=\lim\limits_{n\to\infty}\lambda_{n,u}$, $u\in\langle X\rangle$.
Since $\lambda_{n,u}\in\mathbb{R}$, we conclude that
$\lambda_{u}\in\mathbb{R}$. Thus

$\displaystyle\mathrm{Ln}\,f(t)-it\gamma=\sum_{u\in\langle
X\rangle}\lambda_{u}e^{itu},\quad t\in\mathbb{R},$

with $\lambda_{u}\in\mathbb{R}$ for $u\in\langle X\rangle$, and
$\sum_{u\in\langle X\rangle}|\lambda_{u}|<\infty$. Since
$\bigl{(}\mathrm{Ln}\,f(t)-it\gamma\bigr{)}\bigr{|}_{t=0}=0$, we get
$\lambda_{0}=-\sum_{u\in\langle X\rangle\setminus\\{0\\}}\lambda_{u}$. From
this we obtain (5). $\Box$

##  3 Limit and compactness theorems

In this section we will apply the representations (5) to obtain limit and
compactness theorems with convergence in variation to distribution functions
from $\boldsymbol{D}_{\\!S}$.

Let $(F_{n})_{n\in\mathbb{N}}$ be a sequence of distribution functions and let
$F$ be a distribution function. Recall that, by definition, the sequence
$(F_{n})_{n\in\mathbb{N}}$ converges in variation to $F$ if $\|F_{n}-F\|\to
0$, $n\to\infty$, where $\|\cdot\|$ denotes the total variation on
$\mathbb{R}$. We denote this as $F_{n}\xrightarrow{\mathrm{var}}F$. Here we
are interested in the case, when $F\in\boldsymbol{D}_{\\!S}$. So it is
naturally to consider the following notions of compactness with class
$\boldsymbol{D}_{\\!S}$. We say that $(F_{n})_{n\in\mathbb{N}}$ is
$\boldsymbol{D}_{\\!S}$-relatively compact in variation if every its
subsequence contains a further subsequence converging in variation to a
distribution function from $\boldsymbol{D}_{\\!S}$. We say that
$(F_{n})_{n\in\mathbb{N}}$ is $\boldsymbol{D}_{\\!S}$-stochastically compact
in variation if every its subsequence contains a further subsequence
converging in variation to a nondegenerate distribution function from
$\boldsymbol{D}_{\\!S}$. In particular, if a sequence of distribution
functions converges in variation to some distribution function from
$\boldsymbol{D}_{\\!S}$, then the sequence is
$\boldsymbol{D}_{\\!S}$-relatively compact in variation. If we know that the
limit is nondegenerate, then the sequence is
$\boldsymbol{D}_{\\!S}$-stochastically compact in variation. The introduced
notions of compactness are modified versions of well known relative (weak)
compactness (see [3]) and stochastic compactness (introduced by Feller in
[8]), which are based on weak convergence.

It should be noted that the class $\boldsymbol{D}_{\\!S}$ is not closed under
convergence in variation. Indeed, it is easily seen that the sequence

$\displaystyle
G_{n}(x)\colonequals\bigl{(}\tfrac{1}{2}+\tfrac{1}{2+n}\bigr{)}\,I_{0}(x)+\bigl{(}\tfrac{1}{2}-\tfrac{1}{2+n}\bigr{)}\,I_{1}(x),\quad
x\in\mathbb{R},\quad n\in\mathbb{N},$

converges in variation to the law

$\displaystyle
G(x)\colonequals\tfrac{1}{2}\,I_{0}(x)+\tfrac{1}{2}\,I_{1}(x),\quad
x\in\mathbb{R}.$

However, $G_{n}\in\boldsymbol{D}_{\\!S}$ for every $n\in\mathbb{N}$, but
$G\notin\boldsymbol{D}_{\\!S}$ (see the examples from Section 2). Here we also
see that $(G_{n})_{n\in\mathbb{N}}$ is not $\boldsymbol{D}_{\\!S}$-relatively
(and hence $\boldsymbol{D}_{\\!S}$-stochastically) compact in variation.

We begin with the following observation.

###### Theorem 3

Suppose that $F_{n}\in\boldsymbol{D}$, $n\in\mathbb{N}$. If
$(F_{n})_{n\in\mathbb{N}}$ is $\boldsymbol{D}_{\\!S}$-relatively compact in
variation, then $F_{n}\in\boldsymbol{D}_{\\!S}$ for all sufficiently large
$n\in\mathbb{N}$.

Due to this theorem, the conditions of $\boldsymbol{D}_{\\!S}$-relative
compactness of $F_{n}\in\boldsymbol{D}$, $n\in\mathbb{N}$, can be formulated
in terms of parameters from existing representations of type (5) for
characteristic functions of $F_{n}$ with all sufficiently large
$n\in\mathbb{N}$. In particular, the theorem and this remark are valid for the
sequences from $\boldsymbol{D}$ converging in variation to some
$F\in\boldsymbol{D}_{\\!S}$.

We fix $F\in\boldsymbol{D}_{\\!S}$. Let $f$ be its characteristic function
with representation (5). Without loss of generality we suppose that the whole
sequence $(F_{n})_{n\in\mathbb{N}}$ is from $\boldsymbol{D}_{\\!S}$. Let
$X_{n}$ be the set of all points of growth of $F_{n}$ for every
$n\in\mathbb{N}$. Let $f_{n}$ be the characteristic function of $F_{n}$ for
every $n\in\mathbb{N}$. According to Theorem 2, the functions $f_{n}$,
$n\in\mathbb{N}$, admit representation of type (5). We set

$\displaystyle f_{n}(t)=\exp\biggl{\\{}it\gamma_{n}+\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}(e^{itu}-1)\biggr{\\}},\quad
t\in\mathbb{R},\quad n\in\mathbb{N},$

where $\gamma_{n}\in\langle X_{n}\rangle$, $\lambda_{n,u}\in\mathbb{R}$ for
all $u\in\langle X_{n}\rangle\setminus\\{0\\}$, and $\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}|\lambda_{n,u}|<\infty$.

We adopt the convention:

$\displaystyle\lambda_{u}\colonequals 0,\quad u\notin\langle
X\rangle,\quad\text{and}\quad\lambda_{n,u}\colonequals 0,\quad u\notin\langle
X_{n}\rangle,\quad n\in\mathbb{N}.$ (22)

We propose the following criterion of convergence in variation.

###### Theorem 4

$F_{n}\xrightarrow{\mathrm{var}}F$, $n\to\infty$, if and only if the following
couple of conditions holds:

  * $(i)$

there exists $n_{0}\in\mathbb{N}$ such that $\gamma_{n}=\gamma$, $n\geqslant
n_{0}$;

  * $(ii)$

$\sum\limits_{\begin{subarray}{c}u\in\langle X_{n}\rangle\cup\langle
X\rangle,\\\ u\neq 0\end{subarray}}|\lambda_{n,u}-\lambda_{u}|\to 0$,
$n\to\infty$.

This criterion generalizes the corresponding result from [17] for probability
laws on $\mathbb{Z}$.

We next turn to compactness theorems. We need to introduce the following set

$\displaystyle\mathcal{U}=\bigcup_{n=1}^{\infty}\langle
X_{n}\rangle=\\{u_{0},u_{1},u_{2},\ldots\\}.$

Since always $0\in\langle X_{n}\rangle$, $n\in\mathbb{N}$, we set
$u_{0}\colonequals 0$.

###### Theorem 5

$(F_{n})_{n\in\mathbb{N}}$ is $\boldsymbol{D}_{\\!S}$-relatively compact in
variation if and only if the following ensemble of conditions holds:

  * $(i)$

$\gamma_{n}$, $n\in\mathbb{N}$, have finite number of values from
$\mathcal{U}$;

  * $(ii)$

$\sup\limits_{n\in\mathbb{N}}\sum\limits_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}|\lambda_{n,u}|<\infty$;

  * $(iii)$

$\lim\limits_{N\to\infty}\sup\limits_{n\in\mathbb{N}}\sum\limits_{\begin{subarray}{c}u_{k}\in\,\mathcal{U}\setminus\\{0\\}:\\\
k>N\end{subarray}}|\lambda_{n,u_{k}}|=0$.

###### Theorem 6

$(F_{n})_{n\in\mathbb{N}}$ is $\boldsymbol{D}_{\\!S}$-stochastically compact
in variation if and only if it is $\boldsymbol{D}_{\\!S}$-relatively compact
in variation, and

$\displaystyle\varliminf\limits_{n\to\infty}\sum\limits_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}|\lambda_{n,u}|>0.$ (23)

Theorems 5 and 6 generalize the corresponding results from [11] concerning
compactness for probability laws on $\mathbb{Z}$.

We now turn to the proofs of proposed theorems.

Proof of Theorem 3. Suppose, contrary to our claim, that there exists a
strictly increasing sequence of positive integers $(n_{j})_{j\in\mathbb{N}}$
such that every $F_{n_{j}}$ does not belong to $\boldsymbol{D}_{\\!S}$. Let us
choose from $(F_{n_{j}})_{j\in\mathbb{N}}$ a subsequence
$(F_{n_{j}^{\prime}})_{j\in\mathbb{N}}$ that converges in variation to some
distribution function $F$ from $\boldsymbol{D}_{\\!S}$. Let
$f_{n_{j}^{\prime}}$ be the characteristic function of $F_{n_{j}^{\prime}}$
for every $j\in\mathbb{N}$, and $f$ be the characteristic function of $F$.
Then there exists $\mu>0$ such that $|f(t)|\geqslant\mu>0$ for all
$t\in\mathbb{R}$. Due to the estimate

$\displaystyle\sup_{t\in\mathbb{R}}|f_{n_{j}^{\prime}}(t)-f(t)|=\sup_{t\in\mathbb{R}}\,\biggl{|}\int\limits_{\mathbb{R}}e^{itx}d(F_{n_{j}^{\prime}}(x)-F(x))\biggr{|}\leqslant\|F_{n_{j}^{\prime}}-F\|,\quad
j\in\mathbb{N},$ (24)

we obtain $\sup_{t\in\mathbb{R}}|f_{n_{j}^{\prime}}(t)-f(t)|\to 0$ as
$j\to\infty$. Since

$\displaystyle|f_{n_{j}^{\prime}}(t)|\geqslant|f(t)|-|f_{n_{j}^{\prime}}(t)-f(t)|\geqslant\mu-\sup_{s\in\mathbb{R}}|f_{n_{j}^{\prime}}(s)-f(s)|,\quad
j\in\mathbb{N},$

it is true that $\inf_{t\in\mathbb{R}}|f_{n_{j}^{\prime}}(t)|\geqslant\mu/2>0$
for all sufficiently large $j\in\mathbb{N}$. Hence
$F_{n_{j}^{\prime}}\in\boldsymbol{D}_{\\!S}$ for these $j$, that is a
contradiction. $\Box$

Proof of Theorem 4. We will adopt here the following convention. Let $G$ be a
function of bounded variation on $\mathbb{R}$ with characteristic function
$g$, i.e. $g(t)=\int_{\mathbb{R}}e^{itx}dG(x)$, $t\in\mathbb{R}$. Since the
uniqueness theorem is valid for functions of bounded variation, we set
$\|g\|\colonequals\|G\|$, where the latter is the total variation of $G$ on
$\mathbb{R}$. So $\|g\|=0$ if and only if $g(t)=0$, $t\in\mathbb{R}$. It is
easy to check that $\|c\cdot g\|=|c|\cdot\|g\|$, $c\in\mathbb{R}$. Let $G_{1}$
and $G_{2}$ be functions of bounded variation with characteristic functions
$g_{1}$ and $g_{2}$, correspondingly. From the inequalities
$\|G_{1}+G_{2}\|\leqslant\|G_{1}\|+\|G_{2}\|$ and
$\|G_{1}*G_{2}\|\leqslant\|G_{1}\|\cdot\|G_{2}\|$, we correspondingly have
$\|g_{1}+g_{2}\|\leqslant\|g_{1}\|+\|g_{2}\|$ and $\|g_{1}\cdot
g_{2}\|\leqslant\|g_{1}\|\cdot\|g_{2}\|$. Moreover, such normed space of
functions $g$ will be complete (see [9] p. 165).

Sufficiency. Suppose that $(i)$ and $(ii)$ hold. We define

$\displaystyle\Delta_{n}(t)\colonequals\mathrm{Ln}\,f_{n}(t)-\mathrm{Ln}\,f(t),\quad
t\in\mathbb{R},\quad n\in\mathbb{N}.$ (25)

For $n\geqslant n_{0}$ we have

$\displaystyle\Delta_{n}(t)$ $\displaystyle=$ $\displaystyle\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}(e^{itu}-1)-\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}(e^{itu}-1)$ $\displaystyle=$
$\displaystyle\sum\limits_{\begin{subarray}{c}u\in\langle
X_{n}\rangle\cup\langle X\rangle,\\\ u\neq
0\end{subarray}}(\lambda_{n,u}-\lambda_{u})(e^{itu}-1),\quad t\in\mathbb{R}.$

Define

$\displaystyle\lambda_{n,0}\colonequals-\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u},\quad
n\in\mathbb{N},\quad\text{and}\quad\lambda_{0}\colonequals-\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}.$ (26)

So we have

$\displaystyle\Delta_{n}(t)=\sum\limits_{u\in\langle X_{n}\rangle\cup\langle
X\rangle}(\lambda_{n,u}-\lambda_{u})e^{itu},\quad t\in\mathbb{R},\quad
n\geqslant n_{0}.$ (27)

Therefore

$\displaystyle\|\Delta_{n}\|=\sum\limits_{u\in\langle X_{n}\rangle\cup\langle
X\rangle}|\lambda_{n,u}-\lambda_{u}|,\quad n\geqslant n_{0}.$ (28)

According to (26), observe that for every $n\in\mathbb{N}$

$\displaystyle|\lambda_{n,0}-\lambda_{0}|$ $\displaystyle=$
$\displaystyle\Biggl{|}\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}-\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}\Biggr{|}$
$\displaystyle{}=\Biggl{|}\sum\limits_{\begin{subarray}{c}u\in\langle
X_{n}\rangle\cup\langle X\rangle,\\\ u\neq
0\end{subarray}}(\lambda_{n,u}-\lambda_{u})\Biggr{|}\leqslant\sum\limits_{\begin{subarray}{c}u\in\langle
X_{n}\rangle\cup\langle X\rangle,\\\ u\neq
0\end{subarray}}|\lambda_{n,u}-\lambda_{u}|.$

Then

$\displaystyle\|\Delta_{n}\|=|\lambda_{n,0}-\lambda_{0}|+\sum\limits_{\begin{subarray}{c}u\in\langle
X_{n}\rangle\cup\langle X\rangle,\\\ u\neq
0\end{subarray}}|\lambda_{n,u}-\lambda_{u}|\leqslant
2\cdot\\!\\!\\!\\!\\!\\!\sum\limits_{\begin{subarray}{c}u\in\langle
X_{n}\rangle\cup\langle X\rangle,\\\ u\neq
0\end{subarray}}|\lambda_{n,u}-\lambda_{u}|,\quad n\geqslant n_{0}.$

Due to $(ii)$, we have $\|\Delta_{n}\|\to 0$, $n\to\infty$.

We next write

$\displaystyle
f_{n}(t)-f(t)=f(t)\bigl{(}e^{\Delta_{n}(t)}-1\bigr{)}=f(t)\sum\limits_{k=1}^{\infty}\dfrac{\Delta_{n}(t)^{k}}{k!},\quad
t\in\mathbb{R},\quad n\geqslant n_{0}.$

Hence

$\displaystyle\|F_{n}-F\|=\|f_{n}-f\|=\biggl{\|}f\cdot\sum\limits_{k=1}^{\infty}\dfrac{\Delta_{n}^{k}}{k!}\biggr{\|}\leqslant\|f\|\cdot\biggl{\|}\sum\limits_{k=1}^{\infty}\dfrac{\Delta_{n}^{k}}{k!}\biggr{\|},\quad
n\geqslant n_{0}.$

Here $\|f\|=1$ and

$\displaystyle\biggl{\|}\sum\limits_{k=1}^{\infty}\dfrac{\Delta_{n}^{k}}{k!}\biggr{\|}\leqslant\sum\limits_{k=1}^{\infty}\dfrac{\|\Delta_{n}\|^{k}}{k!}=e^{\|\Delta_{n}\|}-1,\quad
n\geqslant n_{0}.$

Thus $\|F_{n}-F\|\leqslant e^{\|\Delta_{n}\|}-1$, $n\geqslant n_{0}$. Since
$\|\Delta_{n}\|\to 0$ as $n\to\infty$, we obtain $\|F_{n}-F\|\to 0$, i.e.
$F_{n}\xrightarrow{\mathrm{var}}F$.

Necessity. Suppose that $F_{n}\xrightarrow{\mathrm{var}}F$, $n\to\infty$. Then

$\displaystyle\|f_{n}-f\|=\|F_{n}-F\|\to 0,\quad n\to\infty.$ (29)

Due to the estimate $\sup_{t\in\mathbb{R}}|f_{n}(t)-f(t)|\leqslant\|F_{n}-F\|$
(as in (24)), $n\in\mathbb{N}$, we have that

$\displaystyle\sup_{t\in\mathbb{R}}|f_{n}(t)-f(t)|\to 0,\quad n\to\infty.$
(30)

We now consider the functions $\Delta_{n}$, $n\in\mathbb{N}$, defined by (25):

$\displaystyle\Delta_{n}(t)=it(\gamma_{n}-\gamma)+\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\lambda_{n,u}(e^{itu}-1)-\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\lambda_{u}(e^{itu}-1),\quad t\in\mathbb{R},\quad
n\in\mathbb{N}.$

Since for every $n\in\mathbb{N}$ the function $\Delta_{n}$ is continuous and
$\Delta_{n}(0)=0$, it is the distinguished logarithm of the function
$f_{n}/f$. Moreover, for all sufficiently large $n\in\mathbb{N}$

$\displaystyle\Delta_{n}(t)=\ln\bigl{(}f_{n}(t)/f(t)\bigr{)}=\ln\biggl{(}1+\dfrac{f_{n}(t)-f(t)}{f(t)}\biggr{)},\quad
t\in\mathbb{R}.$ (31)

Indeed, by assumption, $f\in\boldsymbol{D}_{\\!S}$, i.e. $f$ satisties $(S)$
with some $\mu>0$. Therefore

$\displaystyle\sup_{t\in\mathbb{R}}\biggl{|}\dfrac{f_{n}(t)-f(t)}{f(t)}\biggr{|}\leqslant\dfrac{\sup_{t\in\mathbb{R}}|f_{n}(t)-f(t)|}{\inf_{t\in\mathbb{R}}|f(t)|}\leqslant\dfrac{1}{\mu}\,\sup_{t\in\mathbb{R}}|f_{n}(t)-f(t)|,\quad
n\in\mathbb{N}.$

Due to (30), the first supremum goes to zero. In particular, there exists
$n_{1}\in\mathbb{N}$ such that

$\displaystyle\sup_{t\in\mathbb{R}}\biggl{|}\dfrac{f_{n}(t)-f(t)}{f(t)}\biggr{|}<1,\quad
n\geqslant n_{1}.$ (32)

Consequently, for $n\geqslant n_{1}$ the function in the right-hand side of
(31) is continuous on $\mathbb{R}$, and it equals zero for $t=0$. By
uniqueness, this function is the distinguished logarithm of $f_{n}/f$, i.e.
(31) is valid.

We first check $(i)$. According to the above remark, from representation (25),
on the one hand, we see that $\sup_{t\in\mathbb{R}}|\Delta_{n}(t)|\to 0$,
$n\to\infty$. On the other hand, we have the lower estimate

$\displaystyle\sup_{t\in\mathbb{R}}|\Delta_{n}(t)|$ $\displaystyle\geqslant$
$\displaystyle\sup_{t\in\mathbb{R}}|t(\gamma_{n}-\gamma)|-\sup_{t\in\mathbb{R}}\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}\bigl{|}\lambda_{n,u}(e^{itu}-1)\bigr{|}-\sup_{t\in\mathbb{R}}\sum_{u\in\langle
X\rangle\setminus\\{0\\}}\bigl{|}\lambda_{u}(e^{itu}-1)\bigr{|}$
$\displaystyle\geqslant$
$\displaystyle\sup_{t\in\mathbb{R}}|t(\gamma_{n}-\gamma)|-2\sum_{u\in\langle
X_{n}\rangle\setminus\\{0\\}}|\lambda_{n,u}|-2\sum_{u\in\langle
X\rangle\setminus\\{0\\}}|\lambda_{u}|,\quad n\in\mathbb{N}.$

Hence if there exists a strongly increasing sequence of numbers
$n_{k}\in\mathbb{N}$, $k\in\mathbb{N}$, such that $\gamma_{n_{k}}-\gamma\neq
0$, $k\in\mathbb{N}$, then
$\sup_{t\in\mathbb{R}}|t(\gamma_{n_{k}}-\gamma)|=|\gamma_{n_{k}}-\gamma|\sup_{t\in\mathbb{R}}|t|=\infty$
and, consequently, $\sup_{t\in\mathbb{R}}|\Delta_{n_{k}}(t)|=\infty$,
$k\in\mathbb{N}$. In this case we have a contradiction. Therefore
$\gamma_{n}=\gamma$ for all $n\in\mathbb{N}$ except a finite number, i.e.
$(i)$ is valid with some $n_{0}\in\mathbb{N}$.

We now check $(ii)$. Let us fix positive integer
$n\geqslant\max\\{n_{0},n_{1}\\}$. Then (27) and (28) are valid for
$\Delta_{n}$, where we use (26). Also, due to (31) and (32), we have

$\displaystyle\Delta_{n}(t)=\ln\biggl{(}1+\dfrac{f_{n}(t)-f(t)}{f(t)}\biggr{)}=\sum\limits_{m=1}^{\infty}\dfrac{(-1)^{m-1}}{m}\biggl{(}\dfrac{f_{n}(t)-f(t)}{f(t)}\biggr{)}^{m},\quad
t\in\mathbb{R}.$

Let us consider the function $(f_{n}-f)/f$. It is evident that $f_{n}-f$ is
almost periodic function with absolutely convergent Fourier series. Since $f$
satisfies $(S)$ the function $1/f$ is also almost periodic (see [5] p. 35 or
[14] p. 21) and it has absolutely convergent Fourier series (see [9] p. 175).
Therefore the function $(f_{n}-f)/f$ is almost periodic with absolutely
convergent Fourier series. For every $m\in\mathbb{N}$ the same is true for the
function $((f_{n}-f)/f)^{m}$, and

$\displaystyle\biggl{\|}\biggl{(}\dfrac{f_{n}-f}{f}\biggr{)}^{m}\biggr{\|}\leqslant\biggl{\|}\dfrac{f_{n}-f}{f}\biggr{\|}^{m}\leqslant\bigl{(}\|f_{n}-f\|\cdot\|1/f\|\bigr{)}^{m}.$

Therefore

$\displaystyle\|\Delta_{n}\|=\biggl{\|}\sum\limits_{m=1}^{\infty}\dfrac{(-1)^{m-1}}{m}\biggl{(}\dfrac{f_{n}-f}{f}\biggr{)}^{m}\biggr{\|}\leqslant\sum\limits_{m=1}^{\infty}\dfrac{1}{m}\cdot\bigl{(}\|f_{n}-f\|\cdot\|1/f\|\bigr{)}^{m}.$

From this and (29) we see that $\|\Delta_{n}\|\to 0$ as $n\to\infty$. Thus we
have

$\displaystyle\|\Delta_{n}\|=\sum\limits_{u\in\langle X_{n}\rangle\cup\langle
X\rangle}|\lambda_{n,u}-\lambda_{u}|\to 0,\quad n\to\infty,$

that yields $(ii)$. $\Box$

Proof of Theorem 5. Due to the convention (22), we rewrite $(ii)$ in the
following form

$\displaystyle\sup\limits_{n\in\mathbb{N}}\sum\limits_{k=1}^{\infty}|\lambda_{n,u_{k}}|<\infty.$

This and $(iii)$ together are equivalent to the relative compactness in
$\ell^{1}$ of the sequence $(V_{n})_{n\in\mathbb{N}}$, where
$V_{n}\colonequals(\lambda_{n,u_{k}})_{k\in\mathbb{N}}$, $n\in\mathbb{N}$,
i.e. that every subsequence of $(V_{n})_{n\in\mathbb{N}}$ contains a further
subsequence $(V_{n_{j}})_{j\in\mathbb{N}}$ such that
$\sum_{k=1}^{\infty}|\lambda_{n_{j},u_{k}}-\lambda_{u_{k}}|\to 0$ as
$j\to\infty$, where $(\lambda_{u_{k}})_{k\in\mathbb{N}}$ is some sequence of
real numbers with $\sum_{k=1}^{\infty}|\lambda_{u_{k}}|<\infty$ (see [23], p.
271 and p. 278).

Suppose that $(i)$ holds and $(V_{n})_{n\in\mathbb{N}}$ is relatively compact
in $\ell^{1}$. We show that $(F_{n})_{n\in\mathbb{N}}$ is
$\boldsymbol{D}_{\\!S}$-relatively compact in variation. Let us choose any
subsequence $(F_{n_{j}})_{j\in\mathbb{N}}$. We first choose from
$(n_{j})_{j\in\mathbb{N}}$ a subsequence $(n_{j}^{\prime})_{j\in\mathbb{N}}$
such that $(V_{n^{\prime}_{j}})_{j\in\mathbb{N}}$ converges in $\ell^{1}$ to
some sequence $V=(\lambda_{u_{k}})_{k\in\mathbb{N}}$, $k\in\mathbb{N}$. Next,
since $\gamma_{n_{j}^{\prime}}$, $j\in\mathbb{N}$, have the finite number of
values from $\mathcal{U}$, we can choose from
$(n_{j}^{\prime})_{j\in\mathbb{N}}$ a subsequence
$(n_{j}^{\prime\prime})_{j\in\mathbb{N}}$ such that
$\gamma_{n_{j}^{\prime\prime}}$ is constant for all $j\in\mathbb{N}$, i.e.
there exists $\gamma\in\mathcal{U}$ such that
$\gamma_{n_{j}^{\prime\prime}}=\gamma$, $j\in\mathbb{N}$. It is easily seen
that $f_{n_{j}^{\prime\prime}}(t)\to f(t)$, $j\to\infty$, where

$\displaystyle
f(t)=\exp\biggl{\\{}it\gamma+\sum_{k=1}^{\infty}\lambda_{u_{k}}(e^{itu_{k}}-1)\biggr{\\}},\quad
t\in\mathbb{R}.$

This function is continuous at $t=0$. By the continuity theorem, $f$ is the
characteristic function for some distribution function $F$. It is discrete
(because there is the absolutely convergent series in the exponent) and its
characteristic function $f$ satisfies $(S)$ (see Section 2). So
$F\in\boldsymbol{D}_{\\!S}$ and, by Theorem 4,
$F_{n_{j}^{\prime\prime}}\xrightarrow{\mathrm{var}}F$, $j\to\infty$. Thus we
chosen from the arbitrary subsequence $(F_{n_{j}})_{j\in\mathbb{N}}$ the
further subsequence $(F_{n_{j}^{\prime\prime}})_{j\in\mathbb{N}}$, which
converges in variation to a distribution function from
$\boldsymbol{D}_{\\!S}$. This yields $\boldsymbol{D}_{\\!S}$-relative
compactness in variation for $(F_{n})_{n\in\mathbb{N}}$.

We now suppose that $(F_{n})_{n\in\mathbb{N}}$ is
$\boldsymbol{D}_{\\!S}$-relatively compact in variation. We show that
$(V_{n})_{n\in\mathbb{N}}$ is relatively compact in $\ell^{1}$. Let us choose
any subsequence $(V_{n_{j}})_{j\in\mathbb{N}}$. We next choose from
$(n_{j})_{j\in\mathbb{N}}$ a subsequence $(n_{j}^{\prime})_{j\in\mathbb{N}}$
such that $F_{n_{j}^{\prime}}\xrightarrow{\mathrm{var}}F$, $j\to\infty$, where
$F\in\boldsymbol{D}_{\\!S}$. Let $f$ be its characteristic function with
representation (5). According to Theorem 4, we have

$\displaystyle\sum\limits_{\begin{subarray}{c}u\in\langle
X_{n_{j}^{\prime}}\rangle\cup\langle X\rangle,\\\ u\neq
0\end{subarray}}|\lambda_{n_{j}^{\prime},u}-\lambda_{u}|\to 0,\quad
j\to\infty.$

Due to the convention (22), we can write this in the following form

$\displaystyle\sum\limits_{\begin{subarray}{c}u\in\mathcal{U}\cup\langle
X\rangle,\\\ u\neq 0\end{subarray}}|\lambda_{n_{j}^{\prime},u}-\lambda_{u}|\to
0,\quad j\to\infty.$

It is clear that if $\langle X\rangle\setminus\mathcal{U}\neq\varnothing$ then
$\lambda_{u,n_{j}^{\prime}}=0$, and hence $\lambda_{u}=0$ for all $u\in\langle
X\rangle\setminus\mathcal{U}$. Therefore we have
$\sum_{u\in\mathcal{U}\setminus\\{0\\}}|\lambda_{n_{j}^{\prime},u}-\lambda_{u}|=\sum_{k=1}^{\infty}|\lambda_{n_{j}^{\prime},u_{k}}-\lambda_{u_{k}}|\to
0$, $j\to\infty$. This proves the relative compactness
$(V_{n})_{n\in\mathbb{N}}$ in $\ell^{1}$.

We now show that $(i)$ holds. Suppose, contrary to our claim, that there
exists subsequence $(\gamma_{n_{j}})_{j\in\mathbb{N}}$ with distinct values.
We choose from $(n_{j})_{j\in\mathbb{N}}$ a subsequence
$(n_{j}^{\prime\prime})_{j\in\mathbb{N}}$ such that
$F_{n_{j}^{\prime\prime}}\xrightarrow{\mathrm{var}}F$, $j\to\infty$ with some
$F\in\boldsymbol{D}_{\\!S}$. According to Theorem 4,
$\gamma_{n_{j}^{\prime\prime}}$ is constant for all sufficiently large
$j\in\mathbb{N}$, a contradiction. $\Box$

Proof of Theorem 6. Sufficiency. Let us choose any subsequence
$(F_{n_{j}})_{j\in\mathbb{N}}$. Since $(F_{n})_{n\in\mathbb{N}}$ is
$\boldsymbol{D}_{\\!S}$-relatively compact in variation, we can choose futher
subsequence $(F_{n_{j}^{\prime}})_{j\in\mathbb{N}}$, which converges in
variation to some distribution function $F\in\boldsymbol{D}_{\\!S}$. Let $f$
be its characteristic function with representation (5). If $F$ is degenerate,
then $f(t)=e^{it\gamma}$, i.e. $\lambda_{u}=0$, $u\in\langle
X\rangle\setminus\\{0\\}$. By Theorem 4, we have that
$\gamma_{n_{j}^{\prime}}=\gamma$ for all sufficiently large $j\in\mathbb{N}$,
and

$\displaystyle\sum_{\begin{subarray}{c}u\in\langle
X_{n_{j}^{\prime}}\rangle\cup\langle X\rangle\\\ u\neq
0\end{subarray}}|\lambda_{n_{j}^{\prime},u}|\to 0,\quad j\to\infty.$

This contradicts (23). Hence $F$ is always nondegenerate distribution
function. Thus we have $\boldsymbol{D}_{\\!S}$-stochastic compactness in
variation of $(F_{n})_{n\in\mathbb{N}}$.

Necessity. Suppose that $(F_{n})_{n\in\mathbb{N}}$ is
$\boldsymbol{D}_{\\!S}$-stochastically compact in variation. Then, by
definition, it is $\boldsymbol{D}_{\\!S}$-relatively compact in variation.
Suppose that (23) is not true, i.e. there exists a sequence
$(n_{j})_{j\in\mathbb{N}}$ such that $\sum_{u\in\langle
X_{n_{j}}\rangle\setminus\\{0\\}}|\lambda_{n_{j},u}|\to 0$, $j\to\infty$. By
Theorem 5, $\gamma_{n_{j}}$, $j\in\mathbb{N}$, have finite number of values.
Let us choose from $(n_{j})_{j\in\mathbb{N}}$ a subsequence
$(n_{j}^{\prime})_{j\in\mathbb{N}}$ such that
$\gamma_{n_{j}^{\prime}}=\gamma$, $j\in\mathbb{N}$, with some
$\gamma\in\mathbb{R}$. Then, according to Theorem 4,
$F_{n_{j}^{\prime}}\xrightarrow{\mathrm{var}}F$, $j\to\infty$, where $F$ is
degenerate distribution function with unique growth point $\gamma$. This
contradicts to $\boldsymbol{D}_{\\!S}$-stochastic compactness in variation for
$(F_{n})_{n\in\mathbb{N}}$. $\Box$

##  4 Acknowledgments

The work of I. A. Alexeev was supported by Ministry of Science and Higher
Education of the Russian Federation, agreement 075-15-2019-1619. The work of
A. A. Khartov was supported by RFBR–DFG grant 20-51-12004.

## References

  * [1] Berger, D., Kutlu, M. and Lindner, A. (2021). On multivariate quasi-infinitely divisible distributions. arXiv:2101.02544.
  * [2] Berger, D. and Lindner, A. (2020). A Cramér–Wold device for infinite divisibility of $\mathbb{Z}^{d}$-valued distributions. arXiv:2011.08530.
  * [3] Billingsley, P. (1999). Convergence of Probability Measures. 2nd ed. New York: Wiley.
  * [4] Blum, J. R. and Rosenblatt M. (1959). On the structure of infinitely divisible distributions. Pacific J. Math. 9, n. 1, 1–8.
  * [5] Bohr, H. (1947). Almost Periodic Functions. New York: Chelsea Publishing Company.
  * [6] Chhaiba, H., Demni, N. and Mouayn, Z. (2016). Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels. J. Math. Phys., 57, 7, 072103.
  * [7] Demni, N. and Mouayn, Z. (2015). Analysis of generalized Poisson distributions associated with higher Landau levels. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18, 4, 1550028.
  * [8] Feller, W. (1967). On regular variation and local limit theorems. Proc. V Berkeley Symp. Math. Stats. Prob. 2, part 1, 373–388.
  * [9] Gelfand, I., Raikov, D. and Shilov, G. (1964). Commutative Normed Rings. New York: Chelsea Publishing Company.
  * [10] Gnedenko, B. V. and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Cambridge: Addison-Wesley.
  * [11] Khartov, A. A. (2019). Compactness criteria for quasi-infinitely divisible distributions on the integers. Stat. Probab. Lett. 153, 1–6.
  * [12] Khartov, A. A. and Alexeev, I. A. (2020). Quasi-infinite divisibility and three-point probability laws. Zapiski Nauchnykh Seminarov POMI 495, 305–316 (in Russian).
  * [13] Khinchine, A. Ya. (1938). Limit laws for sums of independent random variables. Moscow-Leningrad: ONTI, (in Russian).
  * [14] Levitan, B. M. (1953). Almost Periodic Functions. Moskow: GITTL (In Russian).
  * [15] Levitan, B. M. and Zhikov, V.V. (1982). Almost Periodic Functions and Differential Equations. Cambridge: Cambridge University Press.
  * [16] Lindner, A. and Sato, K. (2011). Properties of stationary distributions of a sequence of generalized Ornstein-Uhlenbeck processes.  Math. Nachr. 284, 17–18, 2225–2248.
  * [17] Lindner, A., Pan, L. and Sato, K. (2018). On quasi-infinitely divisible distributions. Trans. Amer. Math. Soc. 370, 8483–8520.
  * [18] Linnik, Yu. V. and Ostrovskii, I. V. (1977). Decomposition of Random Variables and Vectors. Transl. Math. Monog., Vol. 48, Providence: AMS.
  * [19] Lukacs, E. (1970). Characteristic Functions. London: Griffin.
  * [20] Nakamura, T. (2015). A complete Riemann zeta distribution and the Riemann hypothesis. Bernoulli 21, 1, 604–617.
  * [21] Passeggeri, P. (2020). Spectral representation of quasi-infinitely divisible processes. Stoch. Process. Appl. 130, Issue 3, 1735–1791.
  * [22] Rocha-Arteaga, A. and Sato, K. (2019). Topics in Infinitely Divisible Distributions and Lévy Processes. Switzerland AG, Cham: Springer Nature.
  * [23] Smirnov, V. I. (1964). A Course of Higher Mathematics. Volume V, Oxford: Pergamon Press.
  * [24] Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. New York: Marcel Dekker Inc.
  * [25] Zhang, H., Liu, Y. and Li, B. (2014). Notes on discrete compound Poisson model with applications to risk theory. Insurance Math. Econom. 59, 325–336.
  * [26] Zolotarev, V. M. (1986). One-dimensional Stable Distributions. Trans. of Math. Monographs vol. 65, Providence: AMS.
  * [27] Zolotarev, V. M. (1997). Modern Theory of Summation of Random Variables. Utrecht: VSP.


††thanks: These authors contributed equally to this work.

# Introducing the monoclinic polymorph of the Kitaev magnet Na2Co2TeO6

Emilie Dufault∗ Department of Physics, Boston College, Chestnut Hill, MA
02467, USA  Faranak Bahrami∗ Department of Physics, Boston College, Chestnut
Hill, MA 02467, USA  Alenna Streeter  Department of Physics, Boston College,
Chestnut Hill, MA 02467, USA  Xiaohan Yao  Department of Physics, Boston
College, Chestnut Hill, MA 02467, USA  Enrique Gonzalez  Department of
Physics, Boston College, Chestnut Hill, MA 02467, USA  Qiang Zhang  Neutron
Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA
Fazel Tafti  Department of Physics, Boston College, Chestnut Hill, MA 02467,
USA<EMAIL_ADDRESS>
###### Abstract

Recent theoretical studies have suggested that the low-energy Hamiltonian of
honeycomb cobaltate systems could be dominated by anisotropic Kitaev
interactions. Motivated by the theory, a honeycomb layered material Na2Co2TeO6
with a hexagonal unit cell has been studied and found to exhibit
antiferromagnetic (AFM) ordering at 27 K with two spin reorientation
transitions at 15 and 5 K. Here we report a monoclinic polymorph of
Na2Co2TeO6, also with honeycomb layered structure but with a single AFM
transition at 9.6 K and without spin reorientation transitions at lower
temperatures. Using neutron diffraction, we identify an in-plane zigzag AFM
order in the ground-state with the spins canted out of the honeycomb planes
and ferromagnetically coupled between them. The zigzag order is suppressed by
a magnetic field of 6 T.

##  I Introduction

Establishing a quantum spin-liquid (QSL) phase is highly desired in condensed
matter physics, since the non-abelian anyonic excitations of a QSL can be used
as qubits for topological quantum computing Broholm  _et al._ (2020); Motome
and Nasu (2020); Knolle and Moessner (2019); Savary and Balents (2016); Kitaev
(2003); Nayak  _et al._ (2008). One of the most promising proposals for the
QSL phase is the Kitaev model based on anisotropic interactions among
spin-$1/2$ particles on a honeycomb lattice Kitaev (2006). Experimental
efforts to materialize the Kitaev model have been largely focused on honeycomb
layered structures with heavy transition metals such as $\alpha$-Li2IrO3,
Na2IrO3, Li2RhO3, $\alpha$-RuCl3, Cu2IrO3, Ag3LiIr2O6, Ag3LiRh2O6, Cu3LiIr2O6,
and H3LiIr2O6 Singh  _et al._ (2012); Plumb  _et al._ (2014); Mehlawat  _et
al._ (2017); Mazin  _et al._ (2013); Todorova and Jansen (2011); Abramchuk
_et al._ (2017); Bahrami  _et al._ (2019, 2021, 2022a); Roudebush  _et al._
(2016); Kitagawa  _et al._ (2018); Kenney  _et al._ (2019); Wang  _et al._
(2021); de la Torre  _et al._ (2021); Fabbris  _et al._ (2021); Bahrami  _et
al._ (2022b); de la Torre  _et al._ (2023). The choice of $4d$ and $5d$
transition metals (Ru, Rh, Ir) is due to their strong spin-orbit coupling
(SOC) that induces anisotropic interactions among pseudospin-$1/2$
($J_{\text{eff}}=1/2$) spin-orbital states Chaloupka  _et al._ (2010); Takagi
_et al._ (2019); Witczak-Krempa  _et al._ (2014). Such $J_{\text{eff}}=1/2$
Kramers doublets originate from the low-spin configuration
$t^{5}_{2\textrm{g}}\,e^{0}_{\textrm{g}}$ of the $(4,5)d^{5}$ orbitals of
Ru3+, Rh4+, and Ir4+ subjected to octahedral crystal electric field (CEF) Rau
_et al._ (2014).

Recent theoretical studies have suggested that both the anisotropic exchange
interactions and Kramers doublets can also be realized in the high-spin
configuration $t^{5}_{2g}e^{2}_{g}$ of the $3d^{7}$ orbitals of Co2+ and Ni3+
Sano  _et al._ (2018); Liu and Khaliullin (2018); Stavropoulos  _et al._
(2019); Liu  _et al._ (2020). The tantalizing possibility of synthesizing
Kitaev QSL candidate materials with earth-abundant elements (Co and Ni)
instead of precious metals (Ru, Rh, and Ir) led to a surge of activity on such
materials as Na3Co2SbO6 and Na2Co2TeO6 Wong  _et al._ (2016); Viciu  _et al._
(2007); Yan  _et al._ (2019); Bera  _et al._ (2017); Xiao  _et al._ (2019);
Yao and Li (2020); Lin  _et al._ (2021); Mukherjee  _et al._ (2022). In these
compounds, anisotropic interactions stem from a sizable Hund’s coupling in the
$e_{\textrm{g}}$ manifold and enhanced SOC effect of the ligands due to
proximity of oxygen to heavier Sb or Te atoms Stavropoulos  _et al._ (2019).

Figure 1:  (a) The hexagonal polymorph of Na2Co2TeO6 has considerable sodium
deficiency and site disorder between the layers. The yellow and white colors
show Na occupancy and vacancy, respectively. (b) Both hexagonal ($P6_{3}22$)
and monoclinic ($C2/m$) space groups have honeycomb layers. (c) The monoclinic
polymorph has less inter-layer sodium disorder.

Na3Co2SbO6 crystallizes in the monoclinic space group $C2/m$ similar to the
iridates. It shows anti-ferromagnetic (AFM) ordering at $T_{\text{N}}=8.3$ K
with a positive Curie-Weiss temperature $\Theta_{\text{CW}}=+2.2$ K. The
positive $\Theta_{\text{CW}}$ and a weak hysteresis in $M(H)$ at 2 K despite
AFM ordering suggest a competition between ferromagnetic (FM) and AFM
interactions in this material Wong  _et al._ (2016); Viciu  _et al._ (2007);
Yan  _et al._ (2019). Na2Co2TeO6 instead crystallizes in the hexagonal space
group $P6_{3}22$. It undergoes an AFM transition at 27 K followed by two spin
reorientation transitions at 15 and 5 K. The negative
$\Theta_{\text{CW}}=-8.3$ K in polycrystalline samples confirms dominant AFM
interactions, unlike competing FM and AFM interactions found in Na3Co2SbO6
Xiao  _et al._ (2019); Yao and Li (2020); Lin  _et al._ (2021); Mukherjee  _et
al._ (2022).

Both the monoclinic ($C2/m$) unit cell of Na3Co2SbO6 and hexagonal
($P6_{3}22$) unit cell of Na2Co2TeO6 posses sodium disorder between the
honeycomb layers. However, there is more disorder in the hexagonal structure
because it allows for three inter-layer Wyckoff sites unlike the monoclinic
structure with two inter-layer Wyckoff sites. Such disorder in the inter-layer
site occupancy randomizes the position of oxygen atoms and leads to higher
levels of bond randomness within the honeycomb layers and stacking faults
between them (Fig. 1a).

In this article, we introduce a monoclinic polymorph of Na2Co2TeO6 in the
space group $C2/m$, which is structurally similar to Na3Co2SbO6. As shown in
Fig. 1, the two-layer monoclinic polymorph reported here has a smaller amount
of inter-layer sodium disorder than the three-layer hexagonal polymorph Bera
_et al._ (2017); Xiao  _et al._ (2019); Lin  _et al._ (2021); Mukherjee  _et
al._ (2022). Unlike the hexagonal Na2Co2TeO6 that has three transitions at 27,
15, and 5 K, the monoclinic polymorph has a single AFM transition at 9.6 K.
Also, the large splitting between zero-field-cooled (ZFC) and field-cooled
(FC) susceptibility in the hexagonal Na2Co2TeO6, indicative of spin-glass
behavior, is absent in the monoclinic polymorph consistent with lower disorder
levels.

##  II Experimental Methods

Polycrystalline samples of both hexagonal and monoclinic Na2Co2TeO6 were
synthesized via a solid-state reaction. The precursor materials sodium
carbonate (99.5$\%$), cobalt oxide (99.7$\%$), and tellurium oxide (99.99$\%$)
were mixed and reacted according to the following equation.

$3\textrm{Na}_{2}\textrm{CO}_{3}+2\textrm{Co}_{3}\textrm{O}_{4}+3\textrm{TeO}_{2}\rightarrow
3\textrm{Na}_{2}\textrm{Co}_{2}\textrm{TeO}_{6}$ (1)

The mixture was pressed into a 350 mg pellet, wrapped in a gold foil, and
sintered in a capped alumina crucible at 850∘C for 24 h. It was then cooled to
550∘C and quenched in a dry box. The hexagonal polymorph was obtained by
following Eq. 1 strictly, and the monoclinic polymorph was obtained by adding
30% molar excess of Na2CO3. Both polymorphs were fairly stable in air and had
distinguishable colors of purple (monoclinic) and maroon (hexagonal) as shown
in Fig. 2. We also synthesized the non-magnetic analog Na2Zn2TeO6 with a
similar approach (using 50% additional Na2CO3) to subtract the phonon
background from the heat capacity data.

Powder x-ray diffraction (PXRD) measurements were performed using a Bruker D8
ECO instrument with a Cu-K$\alpha$ source. The FullProf suite Rodríguez-
Carvajal (1993) and VESTA software Momma and Izumi (2011) were used for the
Rietveld refinements and crystal visualizations. Magnetization and heat
capacity measurements were performed using a Quantum Design MPMS3 and PPMS
Dynacool, respectively. Neutron powder diffraction (NPD) was performed on the
time-of-flight (TOF) powder diffractometer POWGEN at the Spallation Neutron
Source at Oak Ridge National Laboratory by loading 2.5 g of dried powder into
a vanadium sample can and cooling it in an orange cryostat. For optimal
nuclear and magnetic refinements, two neutron banks with center wavelengths of
1.500 Å and 2.556 Å were selected, respectively, at 100 K and 1.6 K. The
Fullprof $\bm{k}$-Search software was used to identify the magnetic
propagation vector Rodríguez-Carvajal (1993). The Bilbao Crystallographic
Server Perez-Mato  _et al._ (2015) was used for the magnetic symmetry
analysis, and GSAS-II Toby and Von Dreele (2013) was used for the refinements.

##  III Results and Discussion

###  III.1 Structural Analysis

Figure 2:  (a) Rietveld refinement of the PXRD pattern of monoclinic
Na2Co2TeO6. The inset compares PXRD patterns of the monoclinic ($C2/m$) and
hexagonal ($P6_{3}22$) polymorphs. (b) Rietveld refinement of the NPD pattern
at $T\gg T_{\text{N}}$. The inset compares the colors of the monoclinic
(purple) and hexagonal (maroon) polymorphs.

Figures 2a,b show the PXRD and NPD patterns of the monoclinic polymorph of
Na2Co2TeO6 (red empty circles) with Rietveld refinements in the $C2/m$ space
group (black solid lines). The crystallographic solution confirmed by both
PXRD and NPD is visualized in Figs. 1b,c, and the refinement details are
summarized in Appendix A. The inset of Fig. 2a shows visible differences
between the PXRD patterns of the monoclinic ($C2/m$) and hexagonal
($P6_{3}22$) polymorphs. The first peak for the hexagonal compound is located
at a lower angle compared to that of the monoclinic compound suggesting a
stronger inter-layer connection and smaller inter-layer spacing in the
monoclinic polymorph. The inset of Fig. 2b shows that the two polymorphs have
different colors. As shown in Fig. 1, the amount of Na-deficiency between the
layers of monoclinic Na2Co2TeO6 is significantly less than that of the
hexagonal polymorph – a direct result of the change of space group. Therefore,
structural disorders such as bond randomness within the honeycomb layers and
stacking faults between them are fewer in the newly synthesized monoclinic
polymorph.

###  III.2 Magnetic Characterization

Figure 3:  (a) Magnetic susceptibility per mole Co (black) and inverse
susceptibility (red) plotted as a function of temperature. The filled and
empty circles correspond to zero-field-cooled (ZFC) and field-cooled (FC)
data, respectively. The solid black line is a Curie-Weiss (CW) fit above 250
K. Inset shows the $d\chi/dT$ curve to identify $T_{\text{N}}$ (b) Comparison
between $\chi(T)$ in the monoclinic and hexagonal polymorphs of Na2Co2TeO6.
(c) $\chi(T)$ (and $d\chi/dT$ in the inset) at several fields values. (d)
Magnetization as a function of field at 2 and 150 K. Inset shows a weak
hysteresis at small fields.

The monoclinic polymorph of Na2Co2TeO6 has a single AFM transition
characterized by one peak in the susceptibility data $\chi(T)$ without ZFC/FC
splitting (Fig. 3a). The Néel temperature $T_{\text{N}}=9.6(6)$ K is
determined from the peak in $d\chi/dT$ in the inset of Fig. 3a. A comparison
between the $\chi(T)$ curves of the monoclinic and hexagonal polymporphs is
shown in Fig. 3b. The hexagonal polymorph orders at a higher temperature
$T_{\text{N}}=27$ K with two spin reorientation transitions at 15 and 5 K (the
peak and trough in the ZFC data), which are absent in the monoclinic polymorph
Xiao  _et al._ (2019); Lin  _et al._ (2021); Mukherjee  _et al._ (2022).
Figure 3b also shows the absence (presence) of ZFC/FC splitting in the
monoclinic (hexagonal) polymorph indicating the absence (presence) of spin-
glass behavior consistent with less (more) Na disorder. In Appendix B we show
that a lower quality sample of the monoclinic Na2Co2TeO6 has a lower
transition temperature (5.9 K instead of 9.6 K) with an upturn at around 3 K,
suggesting that the previously reported spin re-orientation transitions in
hexagonal Na2Co2TeO6 Xiao  _et al._ (2019); Yao and Li (2020); Lin  _et al._
(2021); Mukherjee  _et al._ (2022) may be due to an impurity phase of the
monoclinic polymorph.

A Curie-Weiss (CW) analysis, $\chi^{-1}=(T-\Theta_{\text{CW}})/C$, at $T>250$
K in Fig. 3a yields a CW temperature of $\Theta_{\text{CW}}=+10.3$ K and an
effective moment of $\mu_{\text{eff}}=4.83\,\mu_{\text{B}}$. The positive sign
of $\Theta_{\text{CW}}$ in the monoclinic Na2Co2TeO6 indicates the presence of
FM correlations, unlike in the hexagonal Na2Co2TeO6 which has a negative CW
temperature ($\Theta_{\text{CW}}=-8.3$ K). In this regard, the behavior of
monoclinic Na2Co2TeO6 is closer to that of Na3Co2SbO6 with
$\Theta_{\text{CW}}=+2.2$ K and an AFM order at $T_{\text{N}}=8.3$ K. The
similar behavior of these two compounds is likely related to having the same
monoclinic structure ($C2/m$). Table 1 summarizes the magnetic parameters of
these materials.

Table 1: Magnetic properties of Na3Co2SbO6 and the hexagonal and monoclinic polymorphs of Na2Co2TeO6.  Material | Na3Co2SbO6 | Na2Co2TeO6 | Na2Co2TeO6
---|---|---|---
| Monoclinic | Hexagonal | Monoclinic
Space group | $C2/m$ | $P6_{3}22$ | $C2/m$
T${}_{\text{N}}$ |  $8.3$ K |  $27$ K |  $9.6$ K
$\Theta_{\text{CW}}$ |  $+2.2$ K |  $-8.3$ K |  $+10.3$ K
$\mu_{\text{eff}}$ |  $5.22$ $\mu_{\text{B}}$ | $5.34~{}\mu_{\text{B}}$ | $4.83~{}\mu_{\text{B}}$
Sm/Co | $1.47\,R\ln(2)$ | $0.70\,R\ln(2)$ | $0.70\,R\ln(2)$
Reference | Wong  _et al._ (2016) | Yao and Li (2020); Xiao  _et al._ (2019) | [this work]

The effective moment of $4.83\,\mu_{\text{B}}$ in the monoclinic Na2Co2TeO6 is
close to the value $4.73\,\mu_{\text{B}}$ expected from a high-spin $3d^{7}$
system with $S=3/2$ and $L_{\text{eff}}=1$ with unquenched orbital moment
($\text{g}=1.6$ instead of $2$). In contrast, the effective moments of
hexagonal Na2Co2TeO6 ($5.34\,\mu_{\text{B}}$) and Na3Co2SbO6
($5.22\,\mu_{\text{B}}$) listed in Table 1 are closer to the value
$5.92\,\mu_{\text{B}}$ expected from a spin-only state with quenched orbital
moment ($\text{g}=2$) Xiao  _et al._ (2019). Thus, the effect of SOC seems to
be stronger in the title compound compared to its counterparts.

Figure 3(c) shows that $T_{\text{N}}$, defined as the peak in $d\chi/dT$, is
suppressed by an external magnetic field of 6 T. A similar behavior is
observed in the hexagonal polymorph, where the suppression of $T_{\text{N}}$
happens at 9 T Lin  _et al._ (2021). Such a behavior is reminiscent of the
field-induced quantum paramagnetic phase proposed for $\alpha$-RuCl3 Banerjee
_et al._ (2018).

Figure 3(d) shows magnetization curves below and above $T_{\text{N}}$ in the
monoclinic Na2Co2TeO6. At 2 K, $M(\mu_{0}H=7~{}\text{T})$ reaches
$1.6\mu_{\text{B}}$, which is close to the local moment found in neutron
diffraction (see below). The inset of Fig. 3(d) shows a weak hysteresis at 2 K
for $H<3$ T, evidence of a finite FM component and competing FM/AFM
interactions. This is consistent with the observed positive
$\Theta_{\text{CW}}$ despite AFM ordering (Table 1) as well as the c-type
zigzag AFM order found by neutron scattering (see below).

###  III.3 Heat Capacity

Figure 4:  (a) Heat capacity divided by temperature ($C/T$) per mole Co or Zn
plotted as a function of temperature for monoclinic Na2Co2TeO6 (red) and
Na2Zn2TeO6(black). The black data is multiplied by 0.95 to correct for the
mass difference between Co and Zn. Inset shows $dC/dT$ at zero field to
determine $T_{\text{N}}$. (b) Magnetic heat capacity ($C_{m}$) and entropy
($S_{m}$) of monoclinic Na2Co2TeO6 in units of $R\ln(2)$ as a function of
temperature. (c) $C/T$ per mole Co as a function of temperature at different
magnetic fields. (d) Suppression of $T_{\text{N}}$ with increasing field
according to $d\chi/dT$ and $dC/dT$ data.

Similar to the magnetic susceptibility data, a single peak is observed at 12 K
in the heat capacity of monoclinic Na2Co2TeO6 due to AFM ordering (Fig. 4a).
The low-temperature spin reorientation transitions found in the hexagonal
Na2Co2TeO6 are absent in the monoclonic polymorph according to both magnetic
susceptibility and heat capacity data (Figs. 3a and 4a). The peak in $dC/dT$
in the inset of Fig. 4a is used to evaluate $T_{\text{N}}=9.6(6)$ K consistent
with the value reported from $d\chi/dT$ in the inset of Fig. 3a. The lower
$T_{\text{N}}$ in the monoclinic polymorph (9.6 K) compared to hexagonal
polymorph (27 K) indicates enhanced magnetic frustration due to the change of
crystal symmetry (Fig. 1).

To isolate the magnetic heat capacity, we synthesized monoclinic Na2Zn2TeO6 (a
non-magnetic analog of the title compound) and measured its purely phononic
heat capacity (black data in Fig. 4a). After subtracting the phonon
background, the magnetic heat capacity ($C_{m}/T$) is plotted in units of
$R\ln(2)$ per mole Co in Fig. 4b (black curve). Also, the magnetic entropy is
calculated by numerical integration using $S_{m}=\int(C_{m}/T)dT$ and plotted
in Fig. 4b (red curve). It reaches $70\%$ of $R\ln(2)$, which is the expected
molar entropy per Co2+ for the theoretically predicted $\Gamma_{7}$ doublet
(pseudo-spin 1/2) Motome  _et al._ (2020). Releasing 70$\%$ of this amount
across the AFM transition could be due to either an incomplete phonon
subtraction or considerable fluctuations of the pseudo-spin $1/2$ degrees of
freedom above $T_{\text{N}}$. Table 1 compares the magnetic entropy of
monoclinic Na2Co2TeO6 with its hexagonal polymorph and the isostructural
system Na3Co2SbO6.

Figure 4c shows that the AFM transition is suppressed gradually by applying a
magnetic field. Using the peaks in both $dC/dT$ and $d\chi/dT$, a temperature-
field phase diagram is constructed in Fig. 4d that shows the suppression of
the AFM order at 6 T. The measured $C/T$ as a function of temperature shows
similar behavior to the magnetic susceptibility and displays a suppression of
the AFM peak with increasing field. However, in contrast to the complete
change of behavior seen in $\chi$ at 6 T, the $C/T$ data still shows a
residual peak up to 9 T. Similar suppression of the AFM peak has been reported
for Na2Co2TeO6 Yao and Li (2020).

###  III.4 Neutron Powder Diffraction

Figure 5:  (a) Neutron powder diffraction (NPD) pattern of the monoclinic
Na2Co2TeO6 at $T<T_{\text{N}}$ modeled by a zigzag magnetic structure
visualized in the bottom panels. (b) The spins are predominantly in the $bc$
plane with $37^{\circ}$ out-of-plane canting. (c) The inter-layer coupling is
FM.

To determine the nuclear and magnetic structures, NPD profiles were collected
at 100 K (Fig. 2b) and 1.6 K (Fig. 5a). The black and red ticks in Fig. 5a
mark the positions of the nuclear and magnetic Bragg peaks, the latter of
which appears at $T<T_{\text{N}}$. The inset of Fig. 5a compares a temperature
independent nuclear Bragg peak at $Q=1.2$ Å to a temperature dependent
magnetic Bragg peak at $Q=0.7$ Å with growing intensity below $T_{\text{N}}$.

The magnetic peaks in Fig. 5a are indexed by the commensurate propagation
vector $\bm{k}=(\frac{1}{2},\frac{1}{2},0)$. A magnetic symmetry analysis
based on the structural space group $C2/m$ gives two magnetic maximal
subgroups corresponding to the zigzag and stripy AFM orders within the
honeycomb layers with FM coupling between the layers. However, the magnetic
refinement for the zigzag order produces a higher quality fit than the stripey
configuration (Appendix C). Thus, the magnetic subgroup that best represents
the experimental data is $P_{s}\overline{1}$ (Irrep: $mV_{1^{-}}$) which
describes a zigzag AFM order within the layers and FM coupling between them
(Figs. 5b,c). The non-vanishing $\chi(T)$ when $T\to 0$ and positive
$\Theta_{\text{CW}}$ in Fig. 3a are consistent with such a magnetic structure.

A refinement of the moment size in the zigzag structure gives
$\mu=(0.48(15),1.50(15),1.18(16))\,\mu_{\text{B}}$ suggesting that the spins
lie primarily along the $b$ axis with $37^{\circ}$ canting out of plane (Figs.
5b,c). The magnetic moment per Co2+ from this refinement is
$1.83~{}\mu_{\text{B}}$ which can be understood by considering the high-spin
configuration (${}^{4}F$) of the $3d^{7}$ orbitals which splits into two
triplets and a singlet (${}^{4}F\to 2^{4}T+^{4}A$) under the octahedral CEF
Lines (1963). The lowest energy triplet ${}^{4}T$ has an orbital angular
momentum $L=-\frac{3}{2}L_{\text{eff}}=-\frac{3}{2}\times 1$ and spin
$\frac{3}{2}$ leading to a total moment $\langle
m\rangle=2S+L=2(\frac{3}{2})-\frac{3}{2}=\frac{3}{2}$. This is close to but
slightly lower than the observed moment of $1.83~{}\mu_{\text{B}}$. The small
difference is likely due to the trigonal distortion which is ignored in the
first-order analysis.

##  IV Conclusion

The results presented here highlight the interplay between the structural
symmetry and magnetic properties in Kitaev magnets. Although both polymorphs
of Na2Co2TeO6 have identical Co-Te honeycomb layers, the magnetic properties
of the monoclinic Na2Co2TeO6 are markedly more similar to the isostructural
Na3Co2SbO6 in space group $C2/m$ than to its hexagonal polymorph in the space
group $P6_{3}22$. Both monoclinic Na2Co2TeO6 and Na3Co2SbO6 have a positive CW
temperature and a single AFM transition with evidence of anisotropic
interactions and magnetic frustration. These results show the importance of
lattice symmetry considerations in the ongoing search for an ideal Kitaev
candidate material.

##  V Acknowledgments

The authors thank S. Nagler and Y. Ran for fruitful discussions. The work at
Boston College was supported by the National Science Foundation under award
number DMR-2203512. A portion of this research used resources at Spallation
Neutron Source, a DOE Office of Science User Facility operated by the Oak
Ridge National Laboratory.

##  Appendix A Rietveld Refinement

Table 2: Unit cell parameters of Na2Co2TeO6 and quality factors of the PXRD Rietveld refinement at room temperature.  Unit cell parameters | Refinement parameters
---|---
Space Group | $C2/m$ | Parameters | 20
$a$ (Å) | $5.33225(6)$ |  $R_{\mathrm{Bragg}}$ (%) | 6.92
$b$ (Å) | $9.20808(8)$ |  $R_{\mathrm{F}}$ (%) | 5.57
$c$ (Å) | $5.80718(8)$ |  $R_{\mathrm{exp}}$ (%) | 5.38
$\beta$ (∘) | $108.90837(88)$ |  $R_{\mathrm{p}}$ (%) | 5.72
V (Å3) | $269.745$ |  $R_{\mathrm{wp}}$ (%) | 7.69
$Z$ | $2$ | $\chi^{2}$ | 2.04
$\rho$ (gr cm-3) | $4.770$ |  $T$ (K) | 295
Table 3: Atomic coordinates, site occupancies, and isotropic Debye-Waller factors from NPD Rietveld refinement of Na2Co2TeO6 in space group $C2/m$ at 100 K.  atom | site | $x$ | $y$ | $z$ | occ. |  $B_{\textrm{iso}}$ (Å2)
---|---|---|---|---|---|---
Na1 | $4h$ | 1/2 | 0.32818 | 1/2 | 0.700 | 0.014
Na2 | $2d$ | 0 | 1/2 | 1/2 | 0.600 | 0.014
Co1 | $4g$ | 0 | 0.66923 | 0 | 1.000 | 0.007
Te1 | $2a$ | 0 | 0 | 0 | 1.000 | 0.0002
O1 | $8j$ | 0.28060 | 0.34569 | 0.80303 | 1.000 | 0.006
O2 | $4i$ | 0.26474 | 1/2 | 0.19231 | 1.000 | 0.006

A co-refinement of PXRD and NPD patterns was used to accurately solve the
crystal structure of monoclinic Na2Co2TeO6. The unit cell parameters from the
PXRD Rietveld refinement are summarized in Table 2. Since neutron diffraction
is more reliable in determining the oxygen positions, the atomic coordinates,
Wyckoff-site occupancies, and Debye-Waller factors are reported from the NPD
refinement in Table 3. Since Na, Co, Te, and O have sufficiently different
atomic form factors for neutron diffraction, the chemical composition of
Na2Co2TeO6 was reliably determined from the NPD refinement.

##  Appendix B Good-quality vs. poor-quality sample

Figure 6:  (a) A good-quality sample (S1) does not have cobalt deficiency
unlike poor-quality (S2) sample. (b) PXRD pattern of the poor-quality sample
(S2) shows a shift of the first peak to the right indicating a larger $c$-axis
due to weaker inter-layer bonding. (c) The magnetic transition is reduced from
9.6 to 5.9 K in the poor-quality sample.

The quality of Na2Co2TeO6 samples varies based on the amount of excess Na2CO3
and the temperature and duration of the synthesis. A common problem in poor-
quality samples is cobalt deficiency that is correlated with excess sodium
between the layers (to maintain charge neutrality). Figures 6a,b show the
results of the PXRD refinements in a good (S1) versus poor (S2) quality
sample. The good quality sample (S1) has less sodium between the layers and no
cobalt deficiency. The poor-quality sample (S2) has more sodium atoms between
the layers which strengthen the inter-layer bonds and shorten the $c$-axis.
Thus, the first Bragg peak in Fig. 6b is shifted to the right in S2 compared
to S1.

Due to cobalt deficiency, $T_{\text{N}}$ is shifted to a lower temperature in
the poor-quality sample (S2) as seen in Fig. 6c. Note that the $T_{\text{N}}$
reduction in S2 is due to disorder; it is not an evidence of increasing
proximity to the Kitaev spin liquid phase. Also, there is an upturn in
$\chi(T)$ of S2 at 3 K similar to the upturn observed in Fig. 3b in the
hexagonal polymorph. It is likely that this upturn is due to disorder (Co
deficiency) in the monoclinic phase and it shows up in hexagonal samples that
are contaminated with a small amount of a parasitic monoclinic phase.

##  Appendix C Neutron Diffraction

Figure 7:  Comparison between the magnetic Rietveld refinements of the NPD
pattern using three different models. (a) Maximal magnetic space group
$P_{s}\overline{1}$ with zigzag order, which gives the best fit and amoment of
$1.83\,\mu_{\text{B}}$ per Co2+. (b) Maximal magnetic space group
$P_{s}\overline{1}$ with stripy order, which gives the worst fit. (c) Magnetic
subgroup $P_{s}1$ with zigzag order but in a lower symmetry magnetic
structure. The fit quality is worse than in panel (a). The red and blue
circles in the insets represent anti-parallel spins. The poor fit quality of
the stripy model leads to a larger weighted profile factor $R_{\text{W}P}$.
The asterisks mark the positions of Co3O4 impurity peaks.

A symmetry analysis of the $\bm{k}=(\frac{1}{2},\frac{1}{2},0)$ wavevector in
the structural space group $C2/m$ of Na2Co2TeO6 gives two magnetic models
belonging to the maximal magnetic space group $P_{s}\overline{1}$. The
irreducible representations of these two magnetic models are $mV_{1}^{-}$ and
$mV_{1}^{+}$ corresponding to the zigzag and stripy orders, respectively. The
Rietveld refinement for both magnetic structures are shown in Fig. 7. Whereas
the zigzag model produces a good fit quality, the stripy model does not fit
the data properly as seen in Fig. 7. For example, the large Bragg peak at
$Q=0.7$ (Å-1) and the small peaks near 1.8 and 1.9 (Å-1) are fitted poorly in
the stripy model. We found a small amount (2% volume fraction) of Co3O4
impurity in our samples. The peaks corresponding to this impurity are marked
by asterisks in Fig. 7.

It is also possible to refine the NPD pattern in a lower symmetry space group
$P_{s}1$ (irrep. $mV_{1}^{+}$) which allows four different Co moment sites,
which we constraint to have the same size. The refinement in this model, which
also gives a zigzag in-plane ordering but with $26^{\circ}$ out-of-plane
canting, is presented in Fig. 7c. This model produces a lower quality fit than
the first zigzag model in Fig. 7a. It also gives a total moment of
$2.91\,\mu_{\text{B}}$ which is considerably higher than the expected moment
from the doublet ground-state ($1.5\,\mu_{\text{B}}$) and should produce twice
the magnetic entropy shown in Fig. 4b. Thus, the model that best describes the
behavior of the title compound is the zigzag model presented in Figs. 5 and
7a.

## References

  * Broholm  _et al._ (2020) C. Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T. Senthil, Science 367, eaay0668 (2020).
  * Motome and Nasu (2020) Y. Motome and J. Nasu, Journal of the Physical Society of Japan 89, 012002 (2020).
  * Knolle and Moessner (2019) J. Knolle and R. Moessner, Annual Review of Condensed Matter Physics 10, 451 (2019).
  * Savary and Balents (2016) L. Savary and L. Balents, Reports on Progress in Physics 80, 016502 (2016).
  * Kitaev (2003) A. Y. Kitaev, Annals of Physics 303, 2 (2003).
  * Nayak  _et al._ (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Reviews of Modern Physics 80, 1083 (2008).
  * Kitaev (2006) A. Kitaev, Annals of Physics 321, 2 (2006), january Special Issue.
  * Singh  _et al._ (2012) Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012).
  * Plumb  _et al._ (2014) K. W. Plumb, J. P. Clancy, L. J. Sandilands, V. V. Shankar, Y. F. Hu, K. S. Burch, H.-Y. Kee, and Y.-J. Kim, Phys. Rev. B 90, 041112 (2014).
  * Mehlawat  _et al._ (2017) K. Mehlawat, A. Thamizhavel, and Y. Singh, Phys. Rev. B 95, 144406 (2017).
  * Mazin  _et al._ (2013) I. I. Mazin, S. Manni, K. Foyevtsova, H. O. Jeschke, P. Gegenwart, and R. Valentí, Phys. Rev. B 88, 035115 (2013).
  * Todorova and Jansen (2011) V. Todorova and M. Jansen, Zeitschrift für anorganische und allgemeine Chemie 637, 37 (2011).
  * Abramchuk  _et al._ (2017) M. Abramchuk, C. Ozsoy-Keskinbora, J. W. Krizan, K. R. Metz, D. C. Bell, and F. Tafti, Journal of the American Chemical Society 139, 15371 (2017).
  * Bahrami  _et al._ (2019) F. Bahrami, W. Lafargue-Dit-Hauret, O. I. Lebedev, R. Movshovich, H.-Y. Yang, D. Broido, X. Rocquefelte, and F. Tafti, Phys. Rev. Lett. 123, 237203 (2019).
  * Bahrami  _et al._ (2021) F. Bahrami, E. M. Kenney, C. Wang, A. Berlie, O. I. Lebedev, M. J. Graf, and F. Tafti, Phys. Rev. B 103, 094427 (2021).
  * Bahrami  _et al._ (2022a) F. Bahrami, X. Hu, Y. Du, O. I. Lebedev, C. Wang, H. Luetkens, G. Fabbris, M. J. Graf, D. Haskel, Y. Ran, and F. Tafti, Science Advances 8, eabl5671 (2022a).
  * Roudebush  _et al._ (2016) J. H. Roudebush, K. Ross, and R. Cava, Dalton Transactions 45, 8783 (2016).
  * Kitagawa  _et al._ (2018) K. Kitagawa, T. Takayama, Y. Matsumoto, A. Kato, R. Takano, Y. Kishimoto, S. Bette, R. Dinnebier, G. Jackeli, and H. Takagi, Nature 554, 341 (2018).
  * Kenney  _et al._ (2019) E. M. Kenney, C. U. Segre, W. Lafargue-Dit-Hauret, O. I. Lebedev, M. Abramchuk, A. Berlie, S. P. Cottrell, G. Simutis, F. Bahrami, N. E. Mordvinova, G. Fabbris, J. L. McChesney, D. Haskel, X. Rocquefelte, M. J. Graf, and F. Tafti, Phys. Rev. B 100, 094418 (2019).
  * Wang  _et al._ (2021) J. Wang, W. Yuan, T. Imai, P. M. Singer, F. Bahrami, and F. Tafti, Phys. Rev. B 103, 214405 (2021).
  * de la Torre  _et al._ (2021) A. de la Torre, B. Zager, F. Bahrami, M. DiScala, J. R. Chamorro, M. H. Upton, G. Fabbris, D. Haskel, D. Casa, T. M. McQueen, F. Tafti, and K. W. Plumb, Phys. Rev. B 104, L100416 (2021).
  * Fabbris  _et al._ (2021) G. Fabbris, A. Thorn, W. Bi, M. Abramchuk, F. Bahrami, J. H. Kim, T. Shinmei, T. Irifune, F. Tafti, A. N. Kolmogorov, and D. Haskel, Phys. Rev. B 104, 014102 (2021).
  * Bahrami  _et al._ (2022b) F. Bahrami, M. Abramchuk, O. Lebedev, and F. Tafti, Molecules 27 (2022b), 10.3390/molecules27030871.
  * de la Torre  _et al._ (2023) A. de la Torre, B. Zager, F. Bahrami, M. H. Upton, J. Kim, G. Fabbris, G.-H. Lee, W. Yang, D. Haskel, F. Tafti, and K. W. Plumb, “Momentum-independent magnetic excitation continuum in the honeycomb iridate H3LiIr2O6,”  (2023), arXiv:2302.07907 [cond-mat].
  * Chaloupka  _et al._ (2010) J. Chaloupka, G. Jackeli, and G. Khaliullin, Physical Review Letters 105, 027204 (2010).
  * Takagi  _et al._ (2019) H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Nature Reviews Physics 1, 264 (2019).
  * Witczak-Krempa  _et al._ (2014) W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, Annual Review of Condensed Matter Physics 5, 57 (2014).
  * Rau  _et al._ (2014) J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Physical Review Letters 112, 077204 (2014).
  * Sano  _et al._ (2018) R. Sano, Y. Kato, and Y. Motome, Phys. Rev. B 97, 014408 (2018).
  * Liu and Khaliullin (2018) H. Liu and G. Khaliullin, Phys. Rev. B 97, 014407 (2018).
  * Stavropoulos  _et al._ (2019) P. P. Stavropoulos, D. Pereira, and H.-Y. Kee, Physical Review Letters 123, 037203 (2019).
  * Liu  _et al._ (2020) H. Liu, J. c. v. Chaloupka, and G. Khaliullin, Phys. Rev. Lett. 125, 047201 (2020).
  * Wong  _et al._ (2016) C. Wong, M. Avdeev, and C. D. Ling, Journal of Solid State Chemistry 243, 18 (2016).
  * Viciu  _et al._ (2007) L. Viciu, Q. Huang, E. Morosan, H. Zandbergen, N. Greenbaum, T. McQueen, and R. Cava, Journal of Solid State Chemistry 180, 1060 (2007).
  * Yan  _et al._ (2019) J.-Q. Yan, S. Okamoto, Y. Wu, Q. Zheng, H. D. Zhou, H. B. Cao, and M. A. McGuire, Physical Review Materials 3, 074405 (2019).
  * Bera  _et al._ (2017) A. K. Bera, S. M. Yusuf, A. Kumar, and C. Ritter, Phys. Rev. B 95, 094424 (2017).
  * Xiao  _et al._ (2019) G. Xiao, Z. Xia, W. Zhang, X. Yue, S. Huang, X. Zhang, F. Yang, Y. Song, M. Wei, H. Deng, _et al._ , Crystal Growth & Design 19, 2658 (2019).
  * Yao and Li (2020) W. Yao and Y. Li, Phys. Rev. B 101, 085120 (2020).
  * Lin  _et al._ (2021) G. Lin, J. Jeong, C. Kim, Y. Wang, Q. Huang, T. Masuda, S. Asai, S. Itoh, G. Günther, M. Russina, _et al._ , Nature communications 12, 5559 (2021).
  * Mukherjee  _et al._ (2022) S. Mukherjee, G. Manna, P. Saha, S. Majumdar, and S. Giri, Phys. Rev. Mater. 6, 054407 (2022).
  * Rodríguez-Carvajal (1993) J. Rodríguez-Carvajal, Physica B: Condensed Matter 192, 55 (1993).
  * Momma and Izumi (2011) K. Momma and F. Izumi, Journal of Applied Crystallography 44, 1272 (2011).
  * Perez-Mato  _et al._ (2015) J. Perez-Mato, S. Gallego, E. Tasci, L. Elcoro, G. de la Flor, and M. Aroyo, Annual Review of Materials Research 45, 217 (2015).
  * Toby and Von Dreele (2013) B. H. Toby and R. B. Von Dreele, Journal of Applied Crystallography 46, 544 (2013).
  * Banerjee  _et al._ (2018) A. Banerjee, P. Lampen-Kelley, J. Knolle, C. Balz, A. A. Aczel, B. Winn, Y. Liu, D. Pajerowski, J. Yan, C. A. Bridges, A. T. Savici, B. C. Chakoumakos, M. D. Lumsden, D. A. Tennant, R. Moessner, D. G. Mandrus, and S. E. Nagler, npj Quantum Materials 3, 1 (2018).
  * Motome  _et al._ (2020) Y. Motome, R. Sano, S. Jang, Y. Sugita, and Y. Kato, Journal of Physics: Condensed Matter 32, 404001 (2020).
  * Lines (1963) M. E. Lines, Physical Review 131, 546 (1963).