# On the Density of Weak Polignac Numbers

Stijn S.C. Hanson<EMAIL_ADDRESS>
###### Abstract.

Let $k$ be an integer which is the difference between prime numbers infinitely
often. It is known that there are infinitely many such $k$ and, in this paper,
we give a new unconditional proof that these $k$ have positive density and
improve on current bounds, assuming a strong hypothesis.

In 2013, Yitang Zhang proved the bounded gap conjecture [5] which asserted
that there exists some natural number $k$ such that there exist infinitely
many pairs of primes whose difference is precisely $k$. We call any $k$ that
satisfies such a property a _weak Polignac number_.

More precisely, we first ask what sorts of finite sets $\mathcal{H}$ we could
construct such that their translates $n+\mathcal{H}$ could be prime infinitely
often. That is to say, we want to create a set
$\mathcal{H}=\\{h_{1},h_{2},\ldots,h_{k}\\}$ where, for all primes $p$, there
is some natural number $m$ such that

$h_{i}\not\equiv m\pmod{p}$

for all $1\leq i\leq k$. Any such $\mathcal{H}$ is called an _admissible set_.

This theorem of Zhang’s (which was later improved upon by James Maynard [1]
and Polymath Project 8 [3] does this by saying that the translates
$n+\mathcal{H}$ of an admissible set $\mathcal{H}$ contain at least two primes
infinitely often. In other words

######  Theorem 1 (Maynard–Polymath–Zhang, 2014).

Suppose that $\mathcal{H}=\\{h_{1},h_{2},\ldots,h_{k}\\}$ is an admissible set
and define the difference set

$\mathcal{D}=\\{h_{j}-h_{i}:h_{i}<h_{j}\\}.$

Then, for $k=50$, $\mathcal{D}$ contains a weak Polignac number.

If we assume stronger conditions such as the Elliott–Halberstam conjecture or
the generalised Elliott–Halberstam conjecture then we get the above theorem
with $k=5$ and $k=3$ respectively.

If we let $\mathcal{P}(x)$ be the number of weak Polignac numbers less than or
equal to $x$ then Pintz has shown [2] that

$\liminf_{x\rightarrow\infty}\frac{\mathcal{P}(x)}{x}>0$

and, in fact, he obtained a positive bound for the lower density. This paper
gives an unconditional bound which, while not being as strong as Pintz’, has a
slightly simpler proof so that, if all we care about is that the lower density
is positive, this method will suffice. We also give a bound which improves
upon Pintz’ under the assumption of the generalised Elliott–Halberstam
conjecture and show that, if it is not optimal, then the methods used can
improve upon the bound by no more than $1/36$.

The method used here is a packing problem: we construct an infinite sequence
of admissible sets $\mathcal{H}^{n}$ such that the difference sets
$\mathcal{D}^{n}$ are disjoint. Previous work in the field has looked for
admissible sets of minimal diameter. These are very useful for lowering the
bound on the Maynard–Polymath–Zhang theorem but are not very suited to this
packing problem as they are rather irregular.

To that end we define a _regular admissible set of size $k$_ to be an
admissible set of the form

$\mathcal{H}_{k}^{n}=\\{0,nP(k),\ldots,(k-1)nP(k)\\}$

where $P(k)=\prod_{p\leq k}p$ is the $k$th primorial and note that their
difference sets are

$\mathcal{D}_{k}^{n}=\\{nP(k),\ldots,(k-1)nP(k)\\}.$

Consider all regular admissible sets of size $k$
$(\mathcal{H}_{k}^{n})_{n\in\mathbb{N}}$. We want to find a subset
$\mathcal{N}\subseteq\mathbb{N}$ such that the sequence
$(\mathcal{D}^{n})_{n\in\mathcal{N}}$ is pairwise disjoint and which pack into
the interval $[1,x]$.

We first note that there can be at most $\frac{x}{(k-1)P(k)}$ different
regular admissible sets which have difference sets which are entirely
contained within $[1,x]$. Then, for $n<m$, $\mathcal{D}_{k}^{n}$ overlaps with
all regular admissible sets $\mathcal{D}_{k}^{m}$ such that

$im=jn$

whenever $1\leq i<j\leq k-1$. There are $\frac{1}{2}(k-1)(k-2)$ different
choices of $i$ and $j$ so every regular admissible set included in our
collection excludes no more than that many other potential admissible sets.
Therefore

$\displaystyle\mathcal{P}(x)$
$\displaystyle\geq\left[\frac{x}{(k-1)P(k)}\right]-\frac{(k-1)(k-2)}{2}\sum_{i=1}^{x}1_{\mathcal{N}}(i)$
$\displaystyle\geq\left[\frac{x}{(k-1)P(k)}\right]-\frac{(k-1)(k-2)}{2}\mathcal{P}(x)$

which implies that.

$\mathcal{P}(x)\geq\frac{2x}{(k-1)((k-1)(k-2)+2)P(k)}+O(1).$

Dividing through by $x$ gives the following result.

######  Theorem 2.

Suppose that all admissible sets of size $k$ contain a weak Polignac number in
their difference sets. Then

$\liminf_{x\rightarrow\infty}\frac{\mathcal{P}(x)}{x}\geq\frac{2}{(k-1)((k-1)(k-2)+2)P(k)}.$

Putting $k=50$ into this formula shows that, unconditionally, the weak
Polignac numbers have density greater than
$\frac{1}{35,462,538,431,226,065,088,930}>2.819\times 10^{-23}$. Then we get a
new derivation of a theorem of Pintz [2]

######  Corollary 3.

The lower asymptotic density of the weak Polignac numbers is positive.

Which, in turn, allows us to apply such wonderful theorems as that of
Szemerédi’s [4] which tells us that there are arbitrarily long arithmetic
progressions within the prime numbers.

Furthermore, under Elliott–Halberstam ($k=5$) the density is greater than
$\frac{1}{840}>0.00119$ and, under Generalised Elliott–Halberstam ($k=3$),
$\frac{1}{24}=0.041\dot{6}$.

Actually, under the Generalised Elliott–Halberstam conjecture we can do a
little better. Let

$\mathcal{A}=\\{6n\leq x-2:n\in\mathbb{N}\\}$

and write $\mathcal{A}=\\{a_{1},a_{2},\ldots,a_{N}\\}$ where $a_{i}=0$
whenever $3\mid i$ and $a_{i}\neq 0$ otherwise. We can also insist that the
non-zero $a_{i}$ are strictly decreasing and define

$\mathcal{H}^{n}=\\{0,2n,2n+a_{n}\\},$

where $3\nmid n$ and $n\leq[x/6]$.

These sets are admissible (as all of their elements are congruent to $0$
modulo $2$ and either $0$ or $2$ modulo $3$). Also all the elements of the
difference sets are contained within $[2,x]$ (as the largest element is
$2+a_{1}\leq x$ by definition). Therefore, in this case,

$\liminf_{x\rightarrow\infty}\frac{\mathcal{P}(x)}{x}\geq\frac{1}{6}.$

We might also like to ask just how successful this stratagem can be. We know
that the $k$-element admissible sets with the smallest difference sets are
regular admissible sets with $k-1$ elements so we can pack no more than

$\left[\frac{x}{2(k-1)}\right]$

even difference sets into $[1,x]$. This gives a maximum packing density of
$\frac{1}{2(k-1)}$ assuming only the bounded gaps conjecture holds for $k$ and
not for $k-1$.

We can improve upon this in particular cases, however. Take $k=3$ and note
that the maximum cardinality for the difference set of an admissible set is
$3$ and the minimum is $2$. Moreover, this minimum is achieved precisely when
the admissible set is regular. We know that regular admissible sets can only
be made up of multiples of $P(k)=6$ and they will have $k-1=2$ elements so
there are no more than

$\left[\frac{x}{12}\right]\sim\frac{x}{12}$

even regular admissible sets with disjoint difference sets contained in
$[1,x]$. We can therefore have an extra

$\left[\frac{\left[\frac{x}{2}\right]-2\left[\frac{x}{12}\right]}{3}\right]\sim\frac{1}{9}$

irregular admissible sets. This tells us that we can have no more than

$\frac{x}{12}+\frac{x}{9}=\frac{7x}{36}$

admissible sets whose difference sets are disjoint and contained in $[1,x]$
for large enough $x$. Therefore, only assuming the generalised
Elliott–Halberstam conjecture, the maximum packing density is $7/36$.

## References

  * [1] Maynard, J. (2015). Small gaps between primes. Annals of Mathematics 181(1); 383-413.
  * [2] Pintz, J. (2002). Polignac Numbers, Conjectures of Erdős on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture. eprint arXiv:1305.6289.
  * [3] Polymath, D. H. J. (2013). A new bound for gaps between primes. Available: http://math.univ-lille1.fr/ bhowmik/seminaire/newgap.pdf.
  * [4] Szemerédi, E. (1975). On sets of integers containing no $k$ elements in arithmetic progressions. Acta Arithmetica 27; 199-245.
  * [5] Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics 179(3); 1121-1174 .


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# On the necessary and sufficient conditions for Peng’s law of large numbers
under sublinear expectations

Xinpeng LI Gaofeng ZONG
Reserach Center for Mathematics and Interdisciplinary Sciences
Shandong University
266237, Qingdao, China;
School of Mathematics and Quantitative Economics,
Shandong University of Finance and Economics,
250014, Jinan, China
∗Email<EMAIL_ADDRESS>Partially supported by NSF of Shandong Province (No.
ZR2021MA018), National Key R&D Program of China (No. 2018YFA0703900), NSF of
China (No.11601281, No.11501325), the Young Scholars Program of Shandong
University, the China Postdoctoral Science Foundation (Grant No.2018T110706,
No.2018M642641)). Corresponding author: Gaofeng Zong, Email<EMAIL_ADDRESS>
Abstract. In this paper, we firstly establish the weak laws of large numbers
on the canonical space
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ by
traditional truncation method and Chebyshev’s inequality as in the classical
probability theory. Then we extend them from the canonical space to the
general sublinear expectation space. The necessary and sufficient conditions
for Peng’s law of large numbers are obtained.
Keywords: Canonical space; Independence and identical distribution; Peng’s law
of large numbers; Sublinear expectation
MSC2010: Primary 60B05; Secondary 60F05, 60A86

##  1 Introduction

The general law of large numbers (LLN) on sublinear expectation space
$(\Omega,\mathcal{H},\mathbb{E})$ was established by Peng [13, 14, 15], known
as Peng’s law of large numbers: Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be
independent and identically distributed (i.i.d.) random variables with

$\lim_{\lambda\to\infty}\mathbb{E}[(|X_{1}|-\lambda)^{+}]=0.$ (1.1)

then

$\lim_{n\to\infty}\mathbb{E}\left[\varphi\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\right)\right]=\max_{-\mathbb{E}[-X_{1}]\leq\mu\leq\mathbb{E}[X_{1}]}\varphi(\mu),\
\ \ \forall\varphi\in C_{b}(\mathbb{R}).$ (1.2)

Here the notions of i.i.d. are initially introduced by Peng [14], which are
directly defined by the sublinear expectation $\mathbb{E}$ instead of the
capacity (for example, see Marinacci [11] and Maccheroni and Marinacci [12]).
The convergence in the form (1.2) is similar to the weak convergence in the
classical probability theory, as such (1.2) can be regarded as the weak LLN.
The limit distribution in the right hand of (1.2) is called maximal
distribution, see Peng [15]. There are also strong LLNs on the sublinear
expectation space, see Chen [1], Chen et al. [3], Zhang [17, 19], Guo and Li
[8], etc.

This paper focuses on the weak LLN on the sublinear expectation space, which
can not be implied directly by the strong one. Recently, the convergence rate
of Peng’s LLN was studied under higher moment conditions, see Fang et al. [4],
Song [16] and Hu et al. [6]. But is (1.1) a sharpest condition to obtain
(1.2)? We recall that the classical weak LLN shows that, for i.i.d. sequence
$\\{X_{i}\\}$ on the probability space $(\Omega,\mathcal{F},P)$,
$\frac{\sum_{i=1}^{n}X_{i}}{n}$ converges to $\mu$ in probability if and only
if

$\displaystyle\lim_{n\rightarrow\infty}nP(|X_{1}|\geq n)=0\ \ \text{and}\ \
\lim_{n\rightarrow\infty}E_{P}[(-n\vee X_{1})\wedge n]=\mu,$ (1.3)

which inspires us to explore the necessary and sufficient conditions for
Peng’s LLN analogous to (1.3).

Different from most literature on the limit theorems under sublinear
expectation, we adopt the formulation similar to Li and Peng [9]. We firstly
establish LLN on the canonical space
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ by
traditional truncation method and Chebyshev’s inequality. Then we extend the
LLN to the general sublinear expectation space introduce by Peng [15]. The
necessary and sufficient conditions for Peng’s LLN are obtained, which is a
natural generalization of the classical LLN to the sublinear expectation
theory. But unlike the classical situation, there is a counterexample shows
that the weak LLN (1.2) does not hold if $\mathbb{E}[|X_{1}|]<\infty$.

The rest of this paper is organized as follows. Section 2 describes the basic
model on the canonical space. We establish the LLNs on the canonical space in
Section 3 and then extend them to the general sublinear expectation space in
Section 4. The proofs are given in Section 5.

##  2 Basic model on canonical space

Following Li and Peng [9], let
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ be the
canonical space and $\\{X_{i}\\}_{i\in\mathbb{N}}$ be the sequence of
canonical random variables defined by $X_{i}(\omega)=\omega_{i}$ for
$\omega=(\omega_{1},\cdots,\omega_{n},\cdots)\in\mathbb{R}^{\mathbb{N}}$. For
each $i\in\mathbb{N}$, $\mathcal{P}_{i}$ is the convex and weakly compact set
of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$
characterizing the uncertainty of the distributions of $X_{i}$ as described in
Peng [15]. The set of joint laws on
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ via the
probability kernels are defined by

$\begin{array}[c]{l}\mathcal{P}=\\{P:P(A^{(n)}\times\mathbb{R}^{\mathbb{N}-n})=\int_{\mathbb{R}}\mu_{1}(dx_{1})\int_{\mathbb{R}}\kappa_{2}(x_{1},dx_{2})\cdots\int_{\mathbb{R}}I_{A^{(n)}}\kappa_{n}(x_{1},\cdots,x_{n-1},dx_{n})\\\
\forall n\geq 1,\ \forall A^{(n)}\in\mathcal{B}(\mathbb{R}^{n}),\ \
\kappa_{i}(x_{1},\cdots,x_{i-1},\cdot)\in\mathcal{P}_{i},\ \ 2\leq i\leq n,\
\mu_{1}\in\mathcal{P}_{1}\\},\end{array}$ (2.1)

where for each $i\geq 2$, $\kappa_{i}(x_{1},\cdots,x_{i-1},dx_{i})$ is the
probability kernel satisfying:

  * (i)

$\forall(x_{1},\cdots,x_{i-1})\in\mathbb{R}^{i-1}$,
$\kappa_{i}(x_{1},\cdots,x_{i-1},\cdot)$ is a probability measure on
$(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

  * (ii)

$\forall B\in\mathcal{B}(\mathbb{R})$, $\kappa_{i}(\cdot,B)$ is
$\mathcal{B}(\mathbb{R}^{i-1})$-measurable.

The existence of $P$ on
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ is shown by
Ionescu-Tulcea theorem.

Since $\mathcal{P}_{i}$ is independent of $(x_{1},\cdots,x_{i-1})$, we call
that $X_{i}$ is independent of $(X_{1},\cdots,X_{i-1})$. Here independence
means the uncertainty of the distributions of $X_{i}$ is independent of random
vector $(X_{1},\cdots,X_{i-1})$. It is obvious that such independence is not
symmetric in general. If we further assume that the uncertainties of each
$X_{i}$ are the same, i.e., $\mathcal{P}_{i}=\mathcal{P}_{1}$, then
$\\{X_{i}\\}_{i\in\mathbb{N}}$ is an i.i.d. sequence. In this case,
$\mathcal{P}$ is determined by the set $\mathcal{P}_{1}$ of the distributions
of $X_{1}$.

For each $\mathcal{B}(\mathbb{R}^{\mathbb{N}})$-measurable random variable
$X$, the sublinear expectation $\mathbb{E}^{\mathcal{P}}$ is defined by

$\mathbb{E}^{\mathcal{P}}[X]:=\sup_{P\in\mathcal{P}}E_{P}[X].$

The corresponding capacities $v$ and $V$ are defined respectively

$v(A)=\inf_{P\in\mathcal{P}}P(A),\ \ \ V(A)=\sup_{P\in\mathcal{P}}P(A),\ \
\forall A\in\mathcal{B}(\mathbb{R}^{\mathbb{N}}).$

The canonical filtration $\\{\mathcal{F}_{i}\\}_{i\in\mathbb{N}}$ is defined
as $\mathcal{F}_{i}=\sigma(X_{k},1\leq k\leq i)$ with convention
$\mathcal{F}_{0}=\\{\emptyset,\mathbb{R}^{\mathbb{N}}\\}$.

By the regularity of conditional probability, we have

$E_{P}[\varphi(X_{i})|\mathcal{F}_{i-1}](\omega)=E_{\kappa_{i}(x_{1},\cdots,x_{i-1},\cdot)}[\varphi(X_{i})],\
\ \ P-\text{a.s.},$

where $\omega=(x_{1},\cdots,x_{n},\cdots)\in\mathbb{R}^{\mathbb{N}}$ and
$\kappa_{i}$ is the probability kernel introduced by $P$.

We immediately have the following proposition since
$\kappa_{i}(x_{1},\cdots,x_{i-1},\cdot)\in\mathcal{P}_{i}$, $1\leq i\leq n$.

######  Proposition 2.1.

For each $P\in\mathcal{P}$ and $\varphi\in C_{b}(\mathbb{R})$, we have

$-\mathbb{E}^{\mathcal{P}}[-\varphi(X_{i})]\leq
E_{P}[\varphi(X_{i})|\mathcal{F}_{i-1}]\leq\mathbb{E}^{\mathcal{P}}[\varphi(X_{i})],\
\ P-\text{a.s.}\ i\in\mathbb{N}.$ (2.2)

######  Remark 2.2.

The inequality (2.2) also holds on the general sublinear expectation space,
see Guo and Li [8].

In the end of this section, we generalize Ottaviani’s inequality to the
canonical space.

######  Proposition 2.3.

Let $c$ be a real number satisfying $0<c<1$ and $S_{n}=\sum_{i=1}^{n}X_{i}$.
If there exist constants $\alpha_{n}\in\mathbb{R}$ such that

$\max_{1\leq k\leq n}V(|S_{n}-S_{k}|\geq\alpha_{n})\leq c,$

then we have

$V\left(\max_{1\leq k\leq n}|S_{k}|\geq
2\alpha_{n}\right)\leq\frac{1}{1-c}V(|S_{n}|\geq\alpha_{n}).$

######  Remark 2.4.

The general Levy maximal inequality on the sublinear expectation space was
firstly proved in Zhang [18], but additional $\varepsilon$-condition is
required.

##  3 Law of large numbers on the canonical space

In this section, we establish the weak LLN on the canonical space by
traditional truncation method and Chebyshev¡¯s inequality as in the classical
probability theory.

Given a weakly compact and convex set $\mathcal{P}_{0}$ of probability
measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, for fixed
$n\in\mathbb{N}$, we can construct $\mathcal{P}$ on
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ by (2.1) with
$\mathcal{P}_{i}=\mathcal{P}_{0}$ for $i\in\mathbb{N}$, such that the sequence
of canonical random variables $\\{X_{i}\\}_{i\in\mathbb{N}}$ are i.i.d., where
$X_{i}(\omega)=\omega_{i}$,
$\forall\omega=(\omega_{1},\cdots,\omega_{n},\cdots)\in\mathbb{R}^{\mathbb{N}}$.
The sublinear expectation $\mathbb{E}^{\mathcal{P}}$ and the capacities $v$
and $V$ corresponding to $\mathcal{P}$ are defined as in the previous section.

Inspired by the weak LLN under sublinear expectation without the moment
condition in Chen et al. [2], we also have the similar LLN on the canonical
space.

######  Theorem 3.1.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a canonical i.i.d. sequence under
$\mathcal{P}$ which is constructed by (2.1), and we further assume that

$\lim_{n\to\infty}nV(|X_{1}|\geq n)=0.$

Then for any $\varepsilon>0$,

$\lim_{n\to\infty}v\left(\underline{\mu}_{n}-\varepsilon\leq\frac{\sum_{i=1}^{n}X_{i}}{n}\leq\overline{\mu}_{n}+\varepsilon\right)=1,$
(3.1)

where $\underline{\mu}_{n}=-\mathbb{E}^{\mathcal{P}}[(-n\vee-X_{1})\wedge n]$
and $\overline{\mu}_{n}=\mathbb{E}^{\mathcal{P}}[(-n\vee X_{1})\wedge n]$.

Furthermore, if $\\{\underline{\mu}_{n}\\}_{n\in\mathbb{N}}$ and
$\\{\overline{\mu}_{n}\\}_{n\in\mathbb{N}}$ are bounded, then

$\lim_{n\to\infty}\left|\mathbb{E}^{\mathcal{P}}\left[\varphi\left(\frac{\sum_{i=1}^{n}X_{i}}{n}\right)\right]-\max_{\underline{\mu}_{n}\leq\mu\leq\overline{\mu}_{n}}\varphi(\mu)\right|=0,\
\ \forall\varphi\in C_{b}(\mathbb{R}).$ (3.2)

Now we establish Peng’s LLN on the canonical space with the necessary and
sufficient conditions.

######  Theorem 3.2.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be a canonical i.i.d. sequence under
$\mathcal{P}$ which is constructed by (2.1). Then

$\lim_{n\to\infty}\mathbb{E}^{\mathcal{P}}\left[\varphi\left(\frac{\sum_{i=1}^{n}X_{i}}{n}\right)\right]=\max_{\underline{\mu}\leq\mu\leq\overline{\mu}}\varphi(\mu),\
\ \forall\varphi\in C_{b}(\mathbb{R}),$ (3.3)

where $-\infty<\underline{\mu}\leq\overline{\mu}<+\infty$, if and only if the
following three conditions hold:

  * (i)

$\lim_{n\to\infty}nV(|X_{1}|\geq n)=0$.

  * (ii)

$\lim_{n\to\infty}\mathbb{E}^{\mathcal{P}}[(-n\vee X_{1})\wedge
n]=\overline{\mu}$.

  * (iii)

$\lim_{n\to\infty}-\mathbb{E}^{\mathcal{P}}[(-n\vee-X_{1})\wedge
n]=\underline{\mu}$.

##  4 Law of large numbers on the sublinear expectation space

In this section, we extend the obtained weak LLN to the general sublinear
expectation space $(\Omega,\mathcal{H},\mathbb{E})$ introduced by Peng [15].

Let $\Omega$ be a given set and let $\mathcal{H}$ be a linear space of real
functions defined on $\Omega$ such that $c\in\mathcal{H}$ for
$c\in\mathbb{R}$, and if $X\in\mathcal{H}$, then $|X|\in\mathcal{H}$. We
further assume that $X_{1},\ldots,X_{n}\in\mathcal{H}$, then
$\varphi(X_{1},\cdots,X_{n})\in\mathcal{H}$ for each $\varphi\in
C_{b.Lip}(\mathbb{R}^{n})$, where $C_{b.Lip}(\mathbb{R}^{n})$ denotes the
space of all bounded and Lipschitz functions on $\mathbb{R}^{n}$.
$\mathcal{H}$ is considered as the space of random variables.

######  Definition 4.1.

A sublinear expectation $\mathbb{E}$ on $\mathcal{H}$ is a functional
$\mathbb{E}:\mathcal{H}\rightarrow\mathbb{R}$ satisfying the following
properties: for all $X,Y\in\mathcal{H}$, we have

(a)

Monotonicity: $\mathbb{E}[X]\geq\mathbb{E}[Y]$ if $X\geq Y$.

(b)

Constant preserving: $\mathbb{E}[c]=c$ for $c\in\mathbb{R}$.

(c)

Sub-additivity: $\mathbb{E}[X+Y]\leq\mathbb{E}[X]+\mathbb{E}[Y]$.

(d)

Positive homogeneity: $\mathbb{E}[\lambda X]=\lambda\mathbb{E}[X]$ for
$\lambda\geq 0$.

The triple $(\Omega,\mathcal{H},\mathbb{E})$ is called a sublinear expectation
space.

Let $X=(X_{1},\cdots,X_{n})$ be a given $n$-dimensional random vector on a
sublinear expectation space $(\Omega,\mathcal{H},\mathbb{E})$. We define a
functional on $C_{b.Lip}(\mathbb{R}^{n})$ by

$\mathbb{{F}}_{X}[\varphi]:=\mathbb{E}[\varphi(X)],\ \ \forall\varphi\in
C_{b.Lip}(\mathbb{R}^{n}).$

The triple
$(\mathbb{R}^{n},C_{b.Lip}(\mathbb{R}^{n}),\mathbb{\hat{F}}_{X}[\cdot])$ forms
a sublinear expectation space, and $\mathbb{\hat{F}}_{X}$ is called the
sublinear distribution of $X$.

######  Definition 4.2.

Let $X$ and $Y$ be two random variables on $(\Omega,\mathcal{H},\mathbb{E})$.
$X$ and $Y$ are called identically distributed, denoted by $X\overset{d}{=}Y$,
if for each $\varphi\in C_{b.Lip}(\mathbb{R})$,

$\mathbb{E}[\varphi(X)]=\mathbb{E}[\varphi(Y)].$

######  Definition 4.3.

Let $\left\\{X_{i}\right\\}_{i\in\mathbb{N}}$ be a sequence of random
variables on $(\Omega,\mathcal{H},\mathbb{E})$. $X_{i}$ is said to be
independent of $\left(X_{1},\ldots,X_{i-1}\right)$ under $\mathbb{E}$, if for
each $\varphi\in C_{b.Lip}(\mathbb{R}^{n})$,

$\mathbb{E}\left[\varphi\left(X_{1},\cdots,X_{i}\right)\right]=\mathbb{E}\left[\left.\mathbb{E}\left[\varphi\left(x_{1},\cdots,x_{i-1},X_{i}\right)\right]\right|_{\left(x_{1},\cdots,x_{i-1}\right)=\left(X_{1},\cdots,X_{i-1}\right)}\right].$

The sequence of random variables $\left\\{X_{i}\right\\}_{i\in\mathbb{N}}$ is
said to be independent, if $X_{i+1}$ is independent of
$\left(X_{1},\ldots,X_{i}\right)$ for each $i\geq 1$.

If the sublinear distribution of $X_{1}$ is given, then there exists a unique
weakly compact and convex set of probability measures $\mathcal{P}_{0}$ on
$(\mathbb{R},\mathcal{B}(\mathbb{R}))$ such that

$\mathbb{E}[\varphi(X_{1})]=\mathbb{E}^{\mathcal{P}_{0}}[\varphi],\
\forall\varphi\in C_{b.Lip}(\mathbb{R}),$ (4.1)

where the existence was proved in Hu and Li [5] and the uniqueness was proved
in Li and Lin [10].

The relation between the i.i.d. sequence on sublinear expectation space and
the one on canonical space is characterized by the following proposition,
which was proved in Li and Peng [9].

######  Proposition 4.4.

Let $\\{X_{i}\\}_{i\in\mathbb{N}}$ be an i.i.d. sequence on sublinear
expectation space $(\Omega,\mathcal{H},\mathbb{E})$ and the sublinear
distribution of $X_{1}$ can be represented by $\mathcal{P}_{0}$ as in (4.1).
Then there exists an sequence of i.i.d. canonical random variables
$\\{Y_{i}\\}_{i\in\mathbb{N}}$ on canonical space
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ with the set
of probability measures $\mathcal{P}$ constructed by (2.1) with
$\mathcal{P}_{i}=\mathcal{P}_{0}$ such that for each $n\in\mathbb{N}$ and
$\varphi\in C_{b.Lip}(\mathbb{R}^{n})$,

$\mathbb{E}[\varphi(X_{1},\cdots,X_{n})]=\mathbb{E}^{\mathcal{P}}[\varphi(Y_{1},\cdots,Y_{n})].$

Now we extend Peng’s LLN (Theorem 3.2) from canonical space to sublinear
expectation space. We need to characterize condition (i) of Theorem 3.2
directly by sublinear expectation $\mathbb{E}$ instead of capacity $V$. For
each $n\in\mathbb{N}$, we define $\psi_{n}\in C_{b.Lip}(\mathbb{R})$ as

$\psi_{n}(x)=\sup_{y\in\mathbb{R}}\\{n\mathbf{1}_{\\{|y|\geq n\\}}-n|y-x|\\}.$

######  Proposition 4.5.

For each $n\in\mathbb{N}$, we have

$n\mathbf{1}_{\\{|x|\geq n\\}}\leq\psi_{n}(x)\leq n\mathbf{1}_{\\{|x|\geq
n-1\\}}.$ (4.2)

######  Theorem 4.6.

Let $\\{X_{i}\\}$ be an i.i.d. sequence on $(\Omega,\mathcal{H},\mathbb{E})$.
Then

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\varphi\left(\frac{\sum_{i=1}^{n}X_{i}}{n}\right)\right]=\max_{\underline{\mu}\leq\mu\leq\overline{\mu}}\varphi(\mu),\
\ \forall\varphi\in C_{b}(\mathbb{R}).$ (4.3)

if and only if the following three conditions hold:

  * (i)

$\lim_{n\rightarrow\infty}\mathbb{E}[\psi_{n}(X_{1})]=0.$

  * (ii)

$\lim_{n\rightarrow\infty}\mathbb{E}[(-n\vee X_{1})\wedge n]=\overline{\mu}$.

  * (iii)

$\lim_{n\rightarrow\infty}-\mathbb{E}[(-n\vee-X_{1})\wedge
n]=\underline{\mu}$.

In the end of this section, we provide some examples. The first one shows that
there exists a random variable $X$ such that
$\lim_{n\to\infty}\mathbb{E}[\psi_{n}(X)]=0$ but condition (1.1) does not
hold, i.e., $\lim_{\lambda\to\infty}\mathbb{E}[(|X|-\lambda)^{+}]\neq 0$.

######  Example 4.7.

Let $\Omega=\mathbb{N}$, $\mathcal{P}=\\{P_{n}:n\in\mathbb{N}\\}$ where
$P_{1}(\\{1\\})=1$ and $P_{n}(\\{1\\})=1-\frac{1}{n^{2}}$,
$P_{n}(\\{kn\\})=\frac{1}{n^{3}}$, $k=1,2,\ldots,n$, for $n=2,3,\cdots$. We
consider a function $X$ on $\mathbb{N}$ defined by $X(n)=n$, $n\in\mathbb{N}$.
We can calculate that
$\lim_{n\to\infty}\mathbb{E}^{\mathcal{P}}[\psi_{n}(X)]=0$ but
$\lim_{\lambda\to\infty}\mathbb{E}^{\mathcal{P}}[(|X|-\lambda)^{+}]=\frac{1}{2}\neq
0$.

The next example shows that the Peng’s LLN may fail only with the first moment
condition, which is different from the classical LLN.

######  Example 4.8.

Let $\Omega=\mathbb{N}$, $\mathcal{P}=\\{P_{n}:n\in\mathbb{N}\\}$, where
$P_{n}(\\{0\\})=1-\frac{1}{n}$ and $P_{n}(\\{n\\})=\frac{1}{n}$ for
$n=1,\cdots,$. Consider a function $X$ on $\mathbb{N}$ defined by $X(n)=n$,
$n\in\mathbb{N}$. The sublinear expectation $\mathbb{E}$ is defined by
$\mathbb{E}[\cdot]=\sup_{P\in\mathcal{P}}E_{P}[\cdot]$. It is clear that
$\mathbb{E}[X]=-\mathbb{E}[-X]=1$. By Peng [15], we can construct i.i.d
sequence $\\{X_{i}\\}_{i\in\mathbb{N}}$ under $\mathbb{E}$ such that $X_{i}$
have the same distribution with $X$. But the weak law of large numbers does
not hold, i.e., we can find $\varphi\in C_{b}(\mathbb{R})$, such that

$\lim_{n\to\infty}\mathbb{E}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]\neq\varphi(1).$

Indeed, we consider $\varphi(x)=1\wedge(1-x)^{+}$. In this case, by simple
calculation, we can obtain
$\mathbb{E}[\varphi(\frac{x+X}{n})]=\varphi(\frac{x}{n}).$ Then we have

$\displaystyle\mathbb{E}\left[\varphi\left(\frac{X_{1}+\cdots+X_{n}}{n}\right)\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\varphi\left(\frac{x+X_{n}}{n}\right)\right]\left|{}_{x=X_{1}+\cdots+X_{n-1}}\right.\right]$
$\displaystyle=\mathbb{E}\left[\varphi\left(\frac{X_{1}+\cdots+X_{n-1}}{n}\right)\right]=\cdots=\mathbb{E}\left[\varphi\left(\frac{X_{1}}{n}\right)\right]=\varphi(0)=1.$

which implies that,

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\varphi\left(\frac{X_{1}+\cdots+X_{n}}{n}\right)\right]=1\neq\varphi(1)=0.$

##  5 Proofs

We firstly prove Ottaviani’s inequality.

###### Proof of Proposition 2.3.

For $k=1,\cdots,n$, let

$A_{k}=\left\\{\max_{l\leq k-1}|S_{l}|<2\alpha_{n},|S_{k}|\geq
2\alpha_{n}\right\\},$

Then

$V\left(\max_{1\leq k\leq n}|S_{k}|\geq
2\alpha_{n}\right)=\mathbb{E}^{\mathcal{P}}[\sum_{k=1}^{n}\mathbf{I}_{A_{k}}].$

We have

$\displaystyle\mathbf{1}_{\\{|S_{n}|\geq\alpha_{n}\\}}$
$\displaystyle\geq\sum_{k=1}^{n}\mathbf{1}_{A_{k}}\mathbf{1}_{\\{|S_{n}|\geq\alpha_{n}\\}}\geq\sum_{k=1}^{n}\mathbf{1}_{A_{k}}\mathbf{1}_{\\{|S_{n}-S_{k}|<\alpha_{n}\\}}$
$\displaystyle=\sum_{k=1}^{n}\mathbf{1}_{A_{k}}(1-\mathbf{1}_{\\{|S_{n}-S_{k}|\geq\alpha_{n}\\}})=(1-c)\sum_{k=1}^{n}\mathbf{1}_{A_{k}}+\sum_{k=1}^{n}\mathbf{1}_{A_{k}}(c-\mathbf{1}_{\\{|S_{n}-S_{k}|\geq\alpha_{n}\\}})$

which implies that

$(1-c)\sum_{k=1}^{n}\mathbf{1}_{A_{k}}\leq\mathbf{1}_{\\{|S_{n}|\geq\alpha_{n}\\}}+\sum_{k=1}^{n}\mathbf{1}_{A_{k}}(\mathbf{1}_{\\{|S_{n}-S_{k}|\geq\alpha_{n}\\}}-c).$

For each $P\in\mathcal{P}$, by (2.1), we obtain

$\displaystyle(1-c)P\left(\max_{k\leq n}|S_{k}|\geq 2\alpha_{n}\right)$
$\displaystyle\leq
P(|S_{n}|\geq\alpha_{n})+\sum_{k=1}^{n}P(A_{k})(P(|S_{n}-S_{k}|\geq\alpha_{n})-c)$
$\displaystyle\leq P(|S_{n}|\geq\alpha_{n})\leq V(|S_{n}|\geq\alpha_{n}).$

The proof is completed. ∎

Now we give the proofs of LLNs on canonical space.

###### Proof of Theorem 3.1.

Let $\tilde{X}_{i}=(-n\vee X_{i})\wedge n$, $1\leq i\leq n$,
$S_{n}=\sum_{i=1}^{n}X_{i}$ and $T_{n}=\sum_{i=1}^{n}\tilde{X}_{i}$.

For each $P\in\mathcal{P}$, Proposition 2.1 implies that

$-\mathbb{E}^{\mathcal{P}}[-\tilde{X}_{i}]\leq
E_{P}[\tilde{X}_{i}|\mathcal{F}_{i-1}]\leq\mathbb{E}^{\mathcal{P}}[\tilde{X}_{i}],\
\ \forall i\in\mathbb{N}.$

By Chebyshev’s inequality, we have

$\displaystyle
P\left(\frac{\sum_{i=1}^{n}\left(\tilde{X}_{i}-E_{P}[\tilde{X}_{i}|\mathcal{F}_{i-1}]\right)}{n}>\varepsilon\right)$
$\displaystyle\leq\frac{\sum_{i=1}^{n}E_{P}\left[\left(\tilde{X}_{i}-E_{P}[\tilde{X}_{i}|\mathcal{F}_{i-1}]\right)^{2}\right]}{n^{2}\varepsilon^{2}}\leq\frac{2\sum_{i=1}^{n}E_{P}[|\tilde{X}_{i}|^{2}]}{n^{2}\varepsilon^{2}}$
$\displaystyle\leq\frac{2\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[|\tilde{X}_{i}|^{2}]}{n^{2}\varepsilon^{2}}=\frac{2\mathbb{E}^{\mathcal{P}}[|\tilde{X}_{1}|^{2}]}{n\varepsilon^{2}}.$

Noting that,

$P(S_{n}\neq T_{n})\leq\sum_{i=1}^{n}P(|X_{i}|\geq
n)\leq\sum_{i=1}^{n}V(|X_{i}|\geq n)=nV(|X_{1}|\geq n),$

thus

$\displaystyle
P\left(\frac{S_{n}}{n}>\overline{\mu}_{n}+\varepsilon\right)\leq$
$\displaystyle P(S_{n}\neq
T_{n})+P\left(\frac{\sum_{i=1}^{n}\left(\tilde{X}_{i}-E_{P}[\tilde{X}_{i}|\mathcal{F}_{i-1}]\right)}{n}>\frac{\varepsilon}{2}\right)$
$\displaystyle+P\left(\frac{\sum_{i=1}^{n}E_{P}[\tilde{X}_{i}|\mathcal{F}_{i-1}]}{n}>\overline{\mu}_{n}+\frac{\varepsilon}{2}\right)$
$\displaystyle\leq$ $\displaystyle nV(|X_{1}|\geq
n)+\frac{8}{n\varepsilon^{2}}\mathbb{E}^{\mathcal{P}}[|\tilde{X}_{1}|^{2}]+P\left(\frac{\sum_{i=1}^{n}\mathbb{E}^{\mathcal{P}}[\tilde{X}_{i}]}{n}>\overline{\mu}_{n}+\frac{\varepsilon}{2}\right)$
$\displaystyle=$ $\displaystyle nV(|X_{1}|\geq
n)+\frac{8}{n\varepsilon^{2}}\mathbb{E}^{\mathcal{P}}[|\tilde{X}_{1}|^{2}]$

We claim that

$\lim_{n\to\infty}\frac{1}{n}\mathbb{E}^{\mathcal{P}}[|\tilde{X}_{1}|^{2}]=0.$

Indeed,

$\displaystyle\frac{1}{n}\mathbb{E}^{\mathcal{P}}[|\tilde{X}_{1}|^{2}]$
$\displaystyle\leq\frac{1}{n}\sup_{P\in\mathcal{P}}E_{P}[|X_{1}|^{2}\mathbf{1}_{\\{|X_{1}|\leq
n\\}}]+nV(|X_{1}|\geq n)$
$\displaystyle\leq\frac{2}{n}\sup_{P\in\mathcal{P}}\int_{0}^{n}tP(|X_{1}|>t)dt+nV(|X_{1}|\geq
n)$ $\displaystyle\leq\frac{2}{n}\int_{0}^{n}tV(|X_{1}|>t)dt+nV(|X_{1}|\geq
n)\to 0,$

since $nV(|X_{1}|\geq n)\to 0$.

Thus we imply

$V\left(\frac{S_{n}}{n}>\overline{\mu}_{n}+\varepsilon\right)\rightarrow 0,$

Similarly, we can prove that
$V(\frac{S_{n}}{n}<\underline{\mu}_{n}-\varepsilon)\rightarrow 0$. Therefore,
(3.1) holds.

The proof of (3.2) is divided into two steps.

Firstly, for each $\varepsilon>0$, we have

$\mathbb{E}^{\mathcal{P}}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]\leq\max_{\underline{\mu}_{n}-\varepsilon\leq\mu\leq\overline{\mu}_{n}+\varepsilon}\varphi(\mu)+\max_{\mu\in\mathbb{R}}\varphi(\mu)\left(V\left(\frac{S_{n}}{n}>\overline{\mu}_{n}+\varepsilon\right)+V\left(\frac{S_{n}}{n}<\underline{\mu}_{n}-\varepsilon\right)\right).$

Let $n\rightarrow\infty$, we obtain

$\limsup_{n\rightarrow\infty}\left\\{\mathbb{E}^{\mathcal{P}}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\max_{\underline{\mu}_{n}-\varepsilon\leq\mu\leq\overline{\mu}_{n}+\varepsilon}\varphi(\mu)\right\\}\leq
0,\ \ \ \forall\varphi\in C_{b}(\mathbb{R}).$

Since $\varphi$ is continuous and $\\{\underline{\mu}_{n}\\}_{n\in\mathbb{N}}$
and $\\{\overline{\mu}_{n}\\}_{n\in\mathbb{N}}$ are bounded, let
$\varepsilon\rightarrow 0$, we imply that,

$\limsup_{n\rightarrow\infty}\left\\{\mathbb{E}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\max_{\underline{\mu}_{n}\leq\mu\leq\overline{\mu}_{n}}\varphi(\mu)\right\\}\leq
0,\ \ \ \forall\varphi\in C_{b}(\mathbb{R}).$ (5.1)

Secondly, for fixed $\varphi\in C_{b}(\mathbb{R})$ with $|\varphi(x)|\leq
C_{\varphi}$, and for each $\tilde{\varepsilon}>0$, there exists
$\mu_{n}\in(\underline{\mu}_{n},\overline{\mu}_{n})$ such that
$\varphi(\mu_{n})\geq\max_{\underline{\mu}_{n}\leq\mu\leq\overline{\mu}_{n}}\varphi(\mu)-\tilde{\varepsilon}$.
By the construction of $\mathcal{P}$, we can find $P_{*}\in\mathcal{P}$ such
that $\\{X_{i}\\}$ is an i.i.d. sequence under $P_{*}$ with
$E_{P_{*}}[\tilde{X}_{1}]=\mu_{n}$. In this case, by the classical weak law of
large numbers, for each $\varepsilon>0$,

$P_{*}\left(\left|\frac{S_{n}}{n}-\mu_{n}\right|>\varepsilon\right)\rightarrow
0.$

By the continuity of $\varphi$, there exists $\delta>0$ such that
$|\varphi(x)-\varphi(y)|<\varepsilon$ provided $|x-y|<\delta$. Then

$\left|E_{P_{*}}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\varphi(\mu_{n})\right|\leq\varepsilon+2C_{\varphi}P_{*}\left(\left|\frac{S_{n}}{n}-\mu_{n}\right|>\delta\right),$

which implies that

$\lim_{n\rightarrow\infty}\left|E_{P_{*}}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\varphi(\mu_{n})\right|=0.$

Finally, for each $\tilde{\varepsilon}>0$,

$\liminf_{n\rightarrow\infty}\left\\{\mathbb{E}^{\mathcal{P}}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\max_{\underline{\mu}_{n}\leq\mu\leq\overline{\mu}_{n}}\varphi(\mu)\right\\}\geq\liminf_{n\rightarrow\infty}\left\\{E_{P_{*}}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\varphi(\mu_{n})\right\\}-\tilde{\varepsilon}=-\tilde{\varepsilon},$
(5.2)

Combining (5.1) and (5.2), we can see that (3.2) holds. ∎

Now we prove the sufficient and necessary conditions for Peng’s LLN on the
canonical space.

###### Proof of Theorem 3.2.

If (i)-(iii) holds, then by Theorem 3.1, (3.3) holds. Conversely, we assume
that (3.3) holds.

Let $\hat{\mu}=2[\overline{\mu}]+2>2\overline{\mu}$, which is an integer. By
(3.3), there exists $N$ such that

$V\left(|S_{n}|\geq\frac{n\hat{\mu}}{2}\right)\leq\frac{1}{4},\ \ \forall
n>N.$

Then for $N\leq k\leq n$, we have

$\displaystyle V(|S_{k}-S_{n}|\geq n\hat{\mu})$ $\displaystyle\leq
V\left(|S_{k}|\geq\frac{n\hat{\mu}}{2}\right)+V\left(|S_{n}|\geq\frac{n\hat{\mu}}{2}\right)$
$\displaystyle\leq
V\left(|S_{k}|\geq\frac{k\hat{\mu}}{2}\right)+V\left(|S_{n}|\geq\frac{n\hat{\mu}}{2}\right)\leq\frac{1}{2}.$

Applying Ottaviani’s inequality, we obtain

$V(\max_{N\leq k\leq n}|S_{k}|\geq 2n\hat{\mu})\leq 2V(|S_{n}|\geq
n\hat{\mu}),$

which implies that

$\displaystyle V(\max_{N+1\leq k\leq n}|X_{k}|\geq 4n\hat{\mu})$
$\displaystyle\leq V(\max_{N+1\leq k\leq n}|S_{k}|\geq
2n\hat{\mu})+V(\max_{N\leq k\leq n-1}|S_{k}|\geq 2n\hat{\mu})$
$\displaystyle\leq 4V(|S_{n}|\geq n\hat{\mu}).$

On the other hand, by the construction of $P$ in (2.1),

$\displaystyle V(\max_{N+1\leq k\leq n}|X_{k}|\geq 4n\hat{\mu})$
$\displaystyle=1-v(\max_{N+1\leq k\leq n}|X_{k}|<4n\hat{\mu})$
$\displaystyle=1-\Pi_{k=N+1}^{n}v(|X_{k}|<4n\hat{\mu})$
$\displaystyle=1-\left(1-V(|X_{1}|\geq 4n\hat{\mu})\right)^{n-N}$
$\displaystyle\geq 1-e^{-(n-N)V(|X_{1}|\geq 4n\hat{\mu})}.$

Thus

$(n-N)V(|X_{1}|\geq 4n\hat{\mu})\leq-\ln(1-4V(|S_{n}|\geq n\hat{\mu})).$

Since $V\left(\frac{|S_{n}|}{n}\geq\hat{\mu}\right)\to 0$, as $n\to\infty$, if
$n>2N$, we have

$\frac{n}{2}V(|X^{n}_{1}|\geq 4n\hat{\mu})\leq(n-N)V(|X_{1}|\geq
4n\hat{\mu})\to 0,$

which implies that

$\lim_{n\to\infty}nV(|X_{1}|\geq n)=0.$

Then by Theorem 3.1, we can deduce that

$\lim_{n\to\infty}v(\underline{\mu}_{n}-\varepsilon\leq\frac{S_{n}}{n}\leq\overline{\mu}_{n}+\varepsilon)=1,\
\ \forall\varepsilon>0,$

where $\overline{\mu}_{n}=\mathbb{E}^{\mathcal{P}}[(-n\vee X_{1})\wedge n]$
and $\underline{\mu}_{n}=-\mathbb{E}^{\mathcal{P}}[(-n\vee-X_{1})\wedge n]$.
Thus

$\lim_{n\to\infty}\left|\mathbb{E}\left[\varphi\left(\frac{S_{n}}{n}\right)\right]-\max_{\underline{\mu}_{n}\leq\mu\leq\overline{\mu}_{n}}\varphi(\mu)\right|=0.$

Combining with (3.3), we can see that (ii) and (iii) hold. ∎

The proof of Proposition 4.5 is just by the simple calculations.

###### Proof of Proposition 4.5.

Taking $y=x$, we immediately obtain

$n\mathbf{1}_{\\{|x|\geq n\\}}\leq\psi_{n}(x).$

We also have

$\displaystyle\psi_{n}(x)=$ $\displaystyle\max\left\\{\sup_{|y-x|\leq
1}\\{n\mathbf{1}_{\\{|y|\geq
n\\}}-n|y-x|\\},\sup_{|y-x|>1}\\{n\mathbf{1}_{\\{|y|\geq
n\\}}-n|y-x|\\}\right\\}$ $\displaystyle\leq$
$\displaystyle\max\\{n\mathbf{1}_{\\{|x|\geq
n-1\\}},0\\}=n\mathbf{1}_{\\{|x|\geq n-1\\}}.$

∎

In the end, we prove the Peng’s LLN on sublinear expectation space with
necessary and sufficient conditions.

###### Proof of Theorem 4.6.

By Proposition 4.4, for each $n\in\mathbb{N}$, there exists a sequence of
i.i.d. canonical random variables $\\{Y_{i}\\}$ on canonical space
$(\mathbb{R}^{\mathbb{N}},\mathcal{B}(\mathbb{R}^{\mathbb{N}}))$ endowed with
the set of probability measures $\mathcal{P}$ constructed by (2.1) with
$\mathcal{P}_{i}=\mathcal{P}_{0}$ such that

$\mathbb{E}\left[\varphi\left(\frac{\sum_{i=1}^{n}X_{i}}{n}\right)\right]=\mathbb{E}^{\mathcal{P}}\left[\varphi\left(\frac{\sum_{i=1}^{n}Y_{i}}{n}\right)\right],\
\ \forall\varphi\in C_{b}(\mathbb{R}).$

Then by Theorem 3.2, (4.3) is equivalent to (i)-(iii) in Theorem 3.2.

It is clear that (ii) and (iii) in Theorem 4.6 is equivalent to (ii) and (iii)
in Theorem 3.2 respectively.

For (i) in Theorem 4.6, it is equivalent to

$\lim_{n\to\infty}\mathbb{E}^{\mathcal{P}}[\psi_{n}(X_{1})]=0,$

which implies that

$nV(|X_{1}|\geq n)\to 0.$

Conversely, by (4.2),

$\mathbb{E}^{\mathcal{P}}[\psi_{n}(X_{1})]\leq nV(|X_{1}|\geq
n-1)=\frac{n}{n-1}\left((n-1)V(|X_{1}|\geq n-1)\right)\to 0.$

Thus (i) in Theorem 3.2 is equivalent to (i) in Theorem 4.6. We complete the
proof. ∎

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DESY 23–146 October 2023
LTH 1353

The double fermionic contribution to the
four-loop quark-to-gluon splitting function
G. Falcioni${}^{\,a,b}$, F. Herzog${}^{\,c}$, S. Moch${}^{\,d}$, J.
Vermaseren${}^{\,e}$ and A. Vogt${}^{\,f}$

aDipartimento di Fisica, Università di Torino, Via Pietro Giuria 1, 10125
Torino, Italy
b Physik-Institut, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich,
Switzerland
cHiggs Centre for Theoretical Physics, School of Physics and Astronomy
The University of Edinburgh, Edinburgh EH9 3FD, Scotland, UK
dII. Institute for Theoretical Physics, Hamburg University
Luruper Chaussee 149, D-22761 Hamburg, Germany
eNikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands

fDepartment of Mathematical Sciences, University of Liverpool
Liverpool L69 3BX, United Kingdom
Abstract

We have computed the first 30 even-$N$ moments for the double fermionic
(${n^{\,2}_{\\!f}}\,$) part of the quark-to-gluon splitting function $P_{\rm
gq}$ at the fourth order of perturbative QCD via the renormalization of off-
shell operator matrix elements. From these results we have determined the
all-$N$ form, and hence the exact $x$-space expression, using systems of
Diophantine equations for its coefficients. The dominant and subdominant
leading small-$x$ ${n^{\,2}_{\\!f}}$ contributions to $P^{\,(3)}_{\rm gq}(x)$
are of the form $x^{\,-1}\ln x$ and $\ln^{\,4\\!}x$, respectively; the leading
large-$x$ term is $\ln^{\,4\\!}{(1\\!-\\!x)}$. The coefficient of the first of
these is new, the other two agree with results obtained before and thus
provide checks of our results.

Next-to-next-to-next-to-leading order (N3LO) calculations of benchmark
processes in perturbative QCD form an important part of the accuracy frontier
at the Large Hadron Collider LHC. Complete analyses at this order require the
4-loop splitting functions for the scale dependence of the proton’s parton
distributions in the fractional momentum $x$, in the flavour singlet sector
given by

$\frac{d}{d\ln{\mu^{\,2}}}\;\Big{(}\begin{array}[]{c}\\!q_{\rm s}\\!\\!\\\
\\!g\\!\end{array}\Big{)}\>=\>\left(\begin{array}[]{cc}\\!P_{\rm qq}&P_{\rm
qg}\\!\\!\\!\\\ \\!P_{\rm gq}&P_{\rm
gg}\\!\\!\\!\end{array}\right)\otimes\Big{(}\begin{array}[]{c}\\!q_{\rm
s}\\!\\!\\\ \\!g\\!\end{array}\Big{)}\quad\mbox{with}\quad P_{\,\rm
ik}(x,\alpha_{\rm s})\,=\,\sum_{n=0}{a_{\rm s}^{\,n+1}}\,P_{\,\rm
ik}^{\,(n)}(x)\>,$ (1)

where $q_{\rm s}\,=\,\sum_{\,i=1}^{\,{n_{\\!f}}}\,(q_{i}+\bar{q}_{i})$ and $g$
are the singlet quark and gluon distributions and $\otimes$ denotes the Mellin
convolution in the momentum variable $x$. The determination of these splitting
functions requires very involved calculations, and only partial results have
been obtained so far.

The leading ${n^{\,3}_{\\!f}}$ contributions in the limit of a large numbers
of light flavours ${n_{\\!f}}$ were completed in ref. [1] together with the
${n^{\,2}_{\\!f}}$ parts of the flavour non-singlet quark-quark splitting
functions $P_{\,\rm ns}^{\,(3)\pm}(x)$. Very recently these results have been
extended to the ${n^{\,2}_{\\!f}}$ part of the pure-singlet quark-quark
splitting function $P_{\,\rm ps}^{\,(3)}(x)\,=\,P_{\,\rm
qq}^{\,(3)}(x)-P_{\,\rm ns}^{\,(3)+}(x)$ [2]. The N3LO non-singlet splitting
functions are completely known in the limit of a large number of colours
${n_{c}}$ [3].

Beyond the quark-quark cases and the ${n^{\,3}_{\\!f}}$ terms, only a limited
number of moments of the 4-loop splitting functions have been computed so far.
These quantities correspond (up to a conventional sign) to the anomalous
dimensions of the gauge invariant operators of twist two,

$\gamma_{\,\rm ik}^{\,(3)}(N)\>=\>-\int_{0}^{1}\\!dx\>\,x^{\,N-1}\,P_{\,\rm
ik}^{\,(3)}(x).$ (2)

Building on recent progress on the renormalization of flavour-singlet operator
matrix elements (OMEs), in particular refs. [4, 5], we have been able to
obtain the even-$N$ Mellin moments to $N=20$ of $P_{\rm ps}^{\,(3)}$ and
$P_{\rm qg}^{\,(3)}$ [6, 7]. For the lower-row quantities in eq. (1), only
determinations to $N\\!=\\!10$ have been completed so far, using the
conceptually much easier but computationally harder route via structure
functions in deep-inelastic scattering (DIS) [8, 9]. In the present letter, we
present the first analytic $x$-dependence of a 4-loop ${n^{\,2}_{\\!f}}$
contribution to eq. (1) beyond the quark-quark case.

As in refs. [6, 7], our calculations are performed in the framework of the
operator-product expansion, via off-shell OME’s. Therefore the renormalization
of the results requires counterterms associated with gauge-variant and ghost
operators, commonly known as aliens. However, only a few alien operators enter
for the ${n^{\,2}_{\\!f}}$ contribution in the quark-to-gluon case. In
particular, this contribution always includes a factor $C_{F}$, hence only the
colour factors ${C^{\,2}_{F}}\,{n^{\,2}_{\\!f}}$ and
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ can appear at four loops. The former
contribution, which is also present in QED, does not require any alien
counterterms. The latter colour factor requires only one class of aliens,
namely (see ref. [6])

$O_{\rm q}^{\,I}\,=\,\eta\,g\,\overline{\psi}\,\hbox to0.0pt{\hbox{$\mskip
1.0mu/$}\hss}\Delta\,t^{a}\,\psi\,\left(\partial^{\,N-2}A_{a}\right)\,,\quad
O_{\rm
g}^{\,I}\,=\,\eta\,\left(D.F\right)^{a}\left(\partial^{\,N-2}A_{a}\right)\,,\quad
O_{\rm
c}^{\,I}\,=\,-\eta\,\left(\partial\,\bar{c}^{\,a}\right)\left(\partial^{\,N-1}c_{a}\right)\,.$
(3)

where $\psi$, $A$ and $c$ represent the quark, gluon and ghost fields, $F$ the
gluon field-strength tensor and $D=\partial-{\rm i}gA$ the covariant
derivative with the coupling $g$, where $g^{2}/(4\pi)=\alpha_{\rm
s}=4\pi\,a_{\rm s}$, after contraction of the Lorentz indices with $N$
identical light-like vectors $\Delta^{\,\mu}$. The mixing $\eta$, which is a
function of $N$ and $\alpha_{\rm s}$, needs to be determined to three loops.
This is achieved efficiently by renormalizing the 3-loop ghost-antighost
correlator with an insertion of the gauge-invariant gluon operators. The
results to $N=60$ agree with all-$N$ the expressions of ref. [5].

The Feynman diagrams contributing to the ${n^{\,2}_{\\!f}}$ part of
$\gamma_{\,\rm gq}^{\,(3)}(N)$ form a small subset of those for the general
case. A few examples are shown in fig. 1. The diagrams are processed in the
usual manner, see ref. [10], by a Form [11, 12, 13] program which collects
self-energy insertions, determines the colour factors [14] and classifies the
topologies according to the conventions of the Forcer program [15]. An
optimized in-house version of this program has been employed to perform the
integral reduction for fixed even values of $N$ in $4-2\varepsilon$
dimensions. For the highest moments of the hardest 3-loop diagrams we have
used the Mincer program [16, 17], suitably extended in its tables from the
version optimized for ref. [18].

Figure 1:  Sample diagrams that enter, after inserting the gluon operator into
one of the gluon lines, the computation of the ${n^{\,2}_{\\!f}}$ part of
$\gamma_{\,\rm gq}^{\,(3)}(N)$ at even values of $N$. Left: the only genuine
4-loop diagram. Due to Furry’s theorem its colour factor is
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$. Right: the 3-loop and (after replacing one of
the gluon lines by a one-loop gluon propagator) 4-loop cases that lead to the
hardest computations at large $N$.

Using these tools, we have been able to compute the anomalous dimensions
$\gamma_{\,\rm gq}^{\,(3)}(N)$ in eq. (2) at all even values $2\leq N\leq 60$.
The corresponding all-$N$ expressions include Riemann-$\zeta$ values, harmonic
sums [19] and simple denominators $\,D_{a}^{\,k}\equiv(N+a)^{-k}$. The range
of $a$, the maximal powers of $D_{a}$ and maximal weights of the sums can be
inferred from the prime-factor decomposition of the denominators at $N\leq
60$. Also lower-order expressions, in particular the
${C^{\,2}_{F}}\,{n_{\\!f}}$ and $C_{F}C_{A}\,{n_{\\!f}}$ parts of
$\gamma_{\,\rm gq}^{\,(2)}(N)$ [20], provide useful information about the
expected analytic form of $\gamma_{\,\rm gq}^{\,(3)}(N)$.

In this manner, we are led to an ansatz of 143 functions for the non-$\zeta$
${C^{\,2}_{F}}\,{n^{\,2}_{\\!f}}$ contribution, hence there are far too many
unknown coefficients for a direct determination from the calculated moments.
However, these coefficients are integer modulo some powers of 2 and 3.
Therefore the resulting system of equations can be turned into a Diophantine
system which requires far fewer equations than unknowns and which can be
addressed by number-theoretical techniques, as, e.g., in refs. [21, 22], based
on the LLL algorithm [23]. This approach has been successfully applied to
3-loop and 4-loop splitting functions before, e.g., in refs. [24, 18, 1, 3].

We have eliminated 19 coefficients – of all functions with overall weights
$w\\!=\\!1$ and $w\\!=\\!2$, and some of $w\\!=\\!3$ – which are mostly
expected to be very large, and determined the remaining 124 unknowns from the
11 equations for $2\leq N\leq 60$, using the program of ref. [22]. 17 of these
124 coefficients turned out to vanish. An analogous procedure was used for the
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ part. We will comment on the differences
between the two cases below.

The resulting expression for the ${n^{\,2}_{\\!f}}$ part of the N3LO quark-to-
gluon splitting function at all even $N\geq 2$ reads

$\displaystyle\gamma_{\rm gq}^{\,(3)}(N)\Big{|}_{{n^{\,2}_{\\!f}}}\;=\;$ (4)
$\displaystyle\mbox{{}}\mbox{\large{$\frac{16}{27}$}}\\*{C^{\,2}_{F}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}{p_{gq}}(N)\\*(-8\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{4}}-16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,3}}-8\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,2}}-8\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{3,1}}+36\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,2}}+28\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2,1}}+16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1,1}})+24\,\\*(N^{2}+N+2)\,\\*({D_{0}}^{2}\,\\*{D_{1}}^{2}\,\\*{D_{2}}\,\\*{D_{-1}^{\,2}})\,\\*(8\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{-3}}-12\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,1}}-12\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{3}}\,\\*(276\,\\*{D_{-1}}-860/3\,\\*{D_{0}}+259/3\,\\*{D_{1}}-64/3\,\\*{D_{2}}-64\,\\*{D_{-1}^{\,2}}-100\,\\*{D_{0}^{\,2}}-48\,\\*{D_{0}^{\,3}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-26\,\\*{D_{1}^{\,2}}+24\,\\*{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2}}\,\\*(-224\,\\*{D_{-1}}+758/3\,\\*{D_{0}}-361/3\,\\*{D_{1}}+16/3\,\\*{D_{2}}+16\,\\*{D_{-1}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+20\,\\*{D_{0}^{\,2}}+24\,\\*{D_{0}^{\,3}}+8\,\\*{D_{1}^{\,2}}-12\,\\*{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1}}\,\\*(-500/3\,\\*{D_{-1}}+586/3\,\\*{D_{0}}-275/3\,\\*{D_{1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+16/3\,\\*{D_{2}}+16\,\\*{D_{-1}^{\,2}}+20\,\\*{D_{0}^{\,2}}+24\,\\*{D_{0}^{\,3}}-12\,\\*{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1}}\,\\*(-148\,\\*{D_{-1}}+364/3\,\\*{D_{0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-409/6\,\\*{D_{1}}-16/3\,\\*{D_{2}}-16\,\\*{D_{-1}^{\,2}}-22\,\\*{D_{0}^{\,2}}-24\,\\*{D_{0}^{\,3}}+45\,\\*{D_{1}^{\,2}}+12\,\\*{D_{1}^{\,3}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{-2}}\,\\*(-5504/9\,\\*{D_{-1}}+504\,\\*{D_{0}}+24\,\\*{D_{1}}+752/9\,\\*{D_{2}}+712/3\,\\*{D_{-1}^{\,2}}+480\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+96\,\\*{D_{0}^{\,3}}+96\,\\*{D_{1}^{\,2}}-144\,\\*{D_{1}^{\,3}}-32/3\,\\*{D_{2}^{\,2}})+{\color[rgb]{1,0,0.87}\mathbf{S}_{2}}\,\\*(5432/9\,\\*{D_{-1}}-1402/3\,\\*{D_{0}}+73/6\,\\*{D_{1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-944/9\,\\*{D_{2}}-416/3\,\\*{D_{-1}^{\,2}}-1600/3\,\\*{D_{0}^{\,2}}-80\,\\*{D_{0}^{\,3}}-120\,\\*{D_{0}^{\,4}}-212/3\,\\*{D_{1}^{\,2}}+130\,\\*{D_{1}^{\,3}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+48\,\\*{D_{1}^{\,4}}-32\,\\*{D_{2}^{\,2}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1}}\,\\*(121/9\,\\*{D_{-1}}+2237/18\,\\*{D_{0}}-2219/36\,\\*{D_{1}}+424/9\,\\*{D_{2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+52\,\\*{D_{-1}^{\,2}}+257/3\,\\*{D_{0}^{\,2}}+102\,\\*{D_{0}^{\,3}}-299/2\,\\*{D_{1}^{\,2}}-20\,\\*{D_{1}^{\,3}}+12\,\\*{D_{1}^{\,4}}+16/3\,\\*{D_{2}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}\,\\*(82801/54\,\\*{D_{-1}}-65503/36\,\\*{D_{0}}+31639/72\,\\*{D_{1}}-3086/27\,\\*{D_{2}}-2660/9\,\\*{D_{-1}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-4915/9\,\\*{D_{0}^{\,2}}-1351\,\\*{D_{0}^{\,3}}+14\,\\*{D_{0}^{\,4}}-312\,\\*{D_{0}^{\,5}}-2219/12\,\\*{D_{1}^{\,2}}-2491/6\,\\*{D_{1}^{\,3}}+85\,\\*{D_{1}^{\,4}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+156\,\\*{D_{1}^{\,5}}-352/3\,\\*{D_{2}^{\,2}}-224/3\,\\*{D_{2}^{\,3}})-1094983/324\,\\*{D_{-1}}+282613/72\,\\*{D_{0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-125953/144\,\\*{D_{1}}+16636/81\,\\*{D_{2}}+19192/27\,\\*{D_{-1}^{\,2}}+1540\,\\*{D_{0}^{\,2}}+2766\,\\*{D_{0}^{\,3}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+515/3\,\\*{D_{0}^{\,4}}+766\,\\*{D_{0}^{\,5}}+2265/8\,\\*{D_{1}^{\,2}}+6425/12\,\\*{D_{1}^{\,3}}-4019/6\,\\*{D_{1}^{\,4}}-256\,\\*{D_{1}^{\,5}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+228\,\\*{D_{1}^{\,6}}+312\,\\*{D_{2}^{\,2}}+1312/9\,\\*{D_{2}^{\,3}}-64\,\\*{D_{2}^{\,4}}\,+\,4\\*{\zeta_{3}}\,\\*(-6\,\\*{p_{gq}}(N)\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}+52\,\\*{D_{-1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-89\,\\*{D_{0}}+76\,\\*{D_{1}}+4\,\\*{D_{2}}+12\,\\*{D_{-1}^{\,2}}+42\,\\*{D_{0}^{\,2}}+9\,\\*{D_{1}^{\,2}})\,-\,72\,\\*{\zeta_{4}}\,\\*{p_{gq}}(N)\bigg{)}$
$\displaystyle\mbox{{}}+\mbox{\large{$\frac{16}{27}$}}\\*{C_{F}}\\*{C_{A}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}-4/3\,\\*{\color[rgb]{1,0,0.87}\theta(N\\!-\\!4)}\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{-2}(N\\!-\\!2)}\,\\*{D_{-2}}+{p_{gq}}(N)\\*(-2\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{-4}}-2\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{4}}-23\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-3,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-20\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,-2}}-2\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,2}}-24\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-3}}+14\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,3}}-12\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,-2}}-8\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,2}}+25\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{3,1}}+20\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,1,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+46\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-2,1}}+58\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,-2}}+27\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,2}}+25\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2,1}}+26\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1,1}}-16\,\\*{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1,1}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{3}}\,\\*(-518/3\,\\*{D_{-1}}+268/3\,\\*{D_{0}}+106/3\,\\*{D_{1}}+40/3\,\\*{D_{2}}+32\,\\*{D_{-1}^{\,2}}+112\,\\*{D_{0}^{\,2}}+58\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{2,1}}\,\\*(-461/3\,\\*{D_{-1}}+427/3\,\\*{D_{0}}-188/3\,\\*{D_{1}}+4/3\,\\*{D_{2}}-16\,\\*{D_{-1}^{\,2}}+28\,\\*{D_{0}^{\,2}}+27\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,2}}\,\\*(-137\,\\*{D_{-1}}+445/3\,\\*{D_{0}}-248/3\,\\*{D_{1}}-4/3\,\\*{D_{2}}-24\,\\*{D_{-1}^{\,2}}+12\,\\*{D_{0}^{\,2}}+5\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1,1}}\,\\*(291\,\\*{D_{-1}}-796/3\,\\*{D_{0}}+299/3\,\\*{D_{1}}-44/3\,\\*{D_{2}}+52\,\\*{D_{-1}^{\,2}}-140\,\\*{D_{0}^{\,2}}-12\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,-2}}\,\\*(-1306/3\,\\*{D_{-1}}+808/3\,\\*{D_{0}}+34/3\,\\*{D_{1}}+16\,\\*{D_{2}}+48\,\\*{D_{-1}^{\,2}}+192\,\\*{D_{0}^{\,2}}+130\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{-2,1}}\,\\*(-914/3\,\\*{D_{-1}}+416/3\,\\*{D_{0}}+242/3\,\\*{D_{1}}+16\,\\*{D_{2}}+48\,\\*{D_{-1}^{\,2}}+192\,\\*{D_{0}^{\,2}}+118\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{-3}}\,\\*(694/3\,\\*{D_{-1}}-404/3\,\\*{D_{0}}-50/3\,\\*{D_{1}}-32/3\,\\*{D_{2}}-32\,\\*{D_{-1}^{\,2}}-112\,\\*{D_{0}^{\,2}}-72\,\\*{D_{1}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{2}}\,\\*(169/6\,\\*{D_{-1}}-59/18\,\\*{D_{0}}+277/18\,\\*{D_{1}}+64/9\,\\*{D_{2}}+284/3\,\\*{D_{-1}^{\,2}}-127/3\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+24\,\\*{D_{0}^{\,3}}-77/2\,\\*{D_{1}^{\,2}}+{D_{1}^{\,3}})+{\color[rgb]{1,0,0.87}{\mathbf{S}}_{1,1}}\,\\*(-23147/36\,\\*{D_{-1}}+19511/36\,\\*{D_{0}}-1630/9\,\\*{D_{1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+74/3\,\\*{D_{2}}-128\,\\*{D_{-1}^{\,2}}+709/2\,\\*{D_{0}^{\,2}}-36\,\\*{D_{0}^{\,3}}+1175/12\,\\*{D_{1}^{\,2}}-28\,\\*{D_{1}^{\,3}}-80/3\,\\*{D_{2}^{\,2}})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}\mathbf{S}_{-2}}\,\\*(5380/9\,\\*{D_{-1}}-1520/3\,\\*{D_{0}}+55/3\,\\*{D_{1}}-376/9\,\\*{D_{2}}-356/3\,\\*{D_{-1}^{\,2}}-1148/3\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-80\,\\*{D_{0}^{\,3}}-422/3\,\\*{D_{1}^{\,2}}+138\,\\*{D_{1}^{\,3}}+16/3\,\\*{D_{2}^{\,2}})+{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}\,\\*(7145/216\,\\*{D_{-1}}+3775/36\,\\*{D_{0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-3527/36\,\\*{D_{1}}+6158/27\,\\*{D_{2}}+1331/6\,\\*{D_{-1}^{\,2}}+4099/36\,\\*{D_{0}^{\,2}}+1523/6\,\\*{D_{0}^{\,3}}+36\,\\*{D_{0}^{\,4}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-128\,\\*{D_{1}^{\,2}}+1513/12\,\\*{D_{1}^{\,3}}-87\,\\*{D_{1}^{\,4}}+1532/9\,\\*{D_{2}^{\,2}}-32/3\,\\*{D_{2}^{\,3}})+304049/324\,\\*{D_{-1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-66833/72\,\\*{D_{0}}+28097/48\,\\*{D_{1}}-57623/81\,\\*{D_{2}}-19435/54\,\\*{D_{-1}^{\,2}}-26555/36\,\\*{D_{0}^{\,2}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-8183/18\,\\*{D_{0}^{\,3}}-356/3\,\\*{D_{0}^{\,4}}-12\,\\*{D_{0}^{\,5}}-17785/72\,\\*{D_{1}^{\,2}}-6941/36\,\\*{D_{1}^{\,3}}+836/3\,\\*{D_{1}^{\,4}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-88\,\\*{D_{1}^{\,5}}-2632/9\,\\*{D_{2}^{\,2}}+256/3\,\\*{D_{2}^{\,3}}\,+\,{\zeta_{3}}\,\\*(\,2\,\\*{\color[rgb]{1,0,0.87}\delta(N\\!-\\!2)}+36\,\\*{p_{gq}}(N)\,\\*{\color[rgb]{1,0,0.87}\mathbf{S}_{1}}-206\,\\*{D_{-1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+378\,\\*{D_{0}}-324\,\\*{D_{1}}-16\,\\*{D_{2}}-48\,\\*{D_{-1}^{\,2}}-168\,\\*{D_{0}^{\,2}}-72\,\\*{D_{1}^{\,2}})+\,72\,\\*{\zeta_{4}}\,\\*{p_{gq}}(N)\bigg{)}\;.$

The ${\zeta_{4}}$ contributions to all 4-loop splitting functions in eq. (1)
have been derived in ref. [25] from the no-$\pi^{2}$ theorem [26, 27]. The
very simple ${\zeta_{4}}$ parts of eq. (4) proportional to

${p_{gq}}(N)\;=\;2/(N-1)\,-\,2/N\,+\,1/(N+1)\;\equiv\;2\,{D_{-1}}-2\,{D_{0}}\,+\,{D_{1}}\;.$
(5)

agree with that result. With the exception of the first term in the
$C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ part, all harmonic sums are to be evaluated at
$N$; this argument has been suppressed in eq. (4) for brevity.

The presence of a contribution with $(N-2)^{-1}\,S_{-2}(N-2)$ – which does, of
course, not lead to a singularity at $N=2$ – is clearly indicated by the
presence of a $\delta(N\\!-\\!2)\,{\zeta_{3}}$ term. Such terms do not occur
in the splitting functions to three loops, but they are present in the
coefficient functions for inclusive DIS already at order ${\alpha_{\rm
s}^{\,2}}$ [28, 29]. Very recently, the same structure was found to appear in
the $C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ part of the pure-singlet anomalous
dimension $\gamma_{\,\rm ps}^{\,(3)}(N)$ [2].

As expected from the values to $N\leq 60$, the overall weight (obtained for
each term by adding the power of $1/(N+a)$ to the weight of the harmonic sum)
reaches $w\\!=\\!6$ for the ${C^{\,2}_{F}}\,{n^{\,2}_{\\!f}}$ part, and
$w\\!=\\!5$ for the $C_{F}C_{A}\,{n^{\,2}_{\\!f}}$ contribution. Only non-
alternating sums of $w\\!=\\!4$ enter the former, as expected from the
${C^{\,2}_{F}}\,{n_{\\!f}}$ 3-loop expression; and the alternating $w\\!=\\!3$
sums occur with a particular common prefactor in the second line of eq. (4).
The endpoint behaviour will be addressed below.

The inverse Mellin transform of eq. (4) can be obtained by an algebraic
procedure [29, 30] based on the fact that harmonic sums occur as coefficients
of the Taylor expansion of harmonic polylogarithms. This procedure results in

$\displaystyle P_{gq}^{\,(3)}(x)\Big{|}_{{n^{\,2}_{\\!f}}}\;=\;$ (6)
$\displaystyle\mbox{{}}\mbox{\large{$\frac{16}{27}$}}\\*{C^{\,2}_{F}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}{(2-x)}\\*\Big{(}156\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{5}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,2}}+60\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{4,0}}+24\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,0,0}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1,0}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-12\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{3}}-156\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0}}-36\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0}}-24\,\\*{\zeta_{4}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{0}}-12\,\\*{\zeta_{2}}\,\\*{\zeta_{3}}-12\,\\*{\zeta_{5}}\Big{)}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{p_{gq}(x)}\\*\Big{(}8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,3}}+28\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,2}}+8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,0}}+16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,1}}-8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0,0}}+16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0,0}}+36\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,1}}-8\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}-28\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}+16\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\Big{)}+{\color[rgb]{1,0,0.87}{\rm{H}}_{4}}\,\\*(14-31\,\\*x)-96\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-3,0}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,2}}\,\\*(-288+96\,\\*x^{-1}+144\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,2}}\,\\*(-20-16\,\\*x^{-1}+28\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,0}}\,\\*(-80+130\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1}}\,\\*(-102+81\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,-1,0}}\,\\*(288-96\,\\*x^{-1}-144\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0,0}}\,\\*(-192+64\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+96\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0,0}}\,\\*(-100-64\,\\*x^{-1}-10\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,0}}\,\\*(-20-16\,\\*x^{-1}+28\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,1}}\,\\*(-22$
$\displaystyle\mbox{{\hskip
8.53581pt}}-16\,\\*x^{-1}+29\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0,0}}\,\\*(-766+485\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3}}\,\\*(1335+390\,\\*x+160/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0}}\,\\*(480$
$\displaystyle\mbox{{\hskip
8.53581pt}}-424/3\,\\*x^{-1}-312\,\\*x+64/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2}}\,\\*(248\,\\*x^{-1}-216\,\\*x+32\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2}}\,\\*(418/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}-332/3\,\\*x^{-1}-275/3\,\\*x+16/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0}}\,\\*(1552/3+368/3\,\\*x^{-1}-149/3\,\\*x+112/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1}}\,\\*(161/3+36\,\\*x^{-1}-244/3\,\\*x+32/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,0}}\,\\*(-248\,\\*x^{-1}+216\,\\*x-32\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0,0}}\,\\*(496/3\,\\*x^{-1}-144\,\\*x+64/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0}}\,\\*(563/3-91/3\,\\*x+160/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0}}\,\\*(764/3-244\,\\*x^{-1}-259/3\,\\*x+64/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0}}\,\\*(542/3-152\,\\*x^{-1}-361/3\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+16/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1}}\,\\*(-268/3+116\,\\*x^{-1}+409/6\,\\*x+16/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2}}\,\\*(-144\,\\*{\zeta_{2}}\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+432\,\\*{\zeta_{2}}-216\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2}}\,\\*(-6727/9-3056/9\,\\*x^{-1}-2891/18\,\\*x-1384/9\,\\*x^{2}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+64\,\\*{\zeta_{2}}\,\\*x^{-1}+164\,\\*{\zeta_{2}}+44\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0}}\,\\*(-160-2408/9\,\\*x^{-1}+56\,\\*x-464/9\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0}}\,\\*(-8110/3-4649/6\,\\*x-2080/9\,\\*x^{2}-14\,\\*{\zeta_{2}}+31\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}\,\\*(-850/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}+3776/9\,\\*x^{-1}+41/6\,\\*x-896/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}\,\\*(-3461/18+491/9\,\\*x^{-1}+2027/36\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}-376/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1}}\,\\*(-372\,\\*{\zeta_{2}}\,\\*x^{-1}+324\,\\*{\zeta_{2}}\,\\*x-48\,\\*{\zeta_{2}}\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0}}\,\\*(1344$
$\displaystyle\mbox{{\hskip
8.53581pt}}+10528/27\,\\*x^{-1}+389/36\,\\*x+6494/27\,\\*x^{2}-424/3\,\\*{\zeta_{2}}\,\\*x^{-1}-1335\,\\*{\zeta_{2}}-702\,\\*{\zeta_{2}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}-160/3\,\\*{\zeta_{2}}\,\\*x^{2}+96\,\\*{\zeta_{3}}\,\\*x^{-1}-48\,\\*{\zeta_{3}}+342\,\\*{\zeta_{3}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\,\\*(72371/36-93103/54\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-30551/72\,\\*x+2678/27\,\\*x^{2}+704/3\,\\*{\zeta_{2}}\,\\*x^{-1}-418/3\,\\*{\zeta_{2}}-49/3\,\\*{\zeta_{2}}\,\\*x-64/3\,\\*{\zeta_{2}}\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}-363779/216+368893/324\,\\*x^{-1}+326947/432\,\\*x-7090/81\,\\*x^{2}+72\,\\*{\zeta_{2}}\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+6727/9\,\\*{\zeta_{2}}+3899/18\,\\*{\zeta_{2}}\,\\*x+1384/9\,\\*{\zeta_{2}}\,\\*x^{2}-116/3\,\\*{\zeta_{3}}\,\\*x^{-1}+674\,\\*{\zeta_{3}}-1288\,\\*{\zeta_{3}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+80/3\,\\*{\zeta_{3}}\,\\*x^{2}+148\,\\*{\zeta_{4}}\,\\*x^{-1}-1229/2\,\\*{\zeta_{4}}+1295/4\,\\*{\zeta_{4}}\,\\*x\bigg{)}$
$\displaystyle\mbox{{}}+\mbox{\large{$\frac{16}{27}$}}\\*{C_{F}}\\*{C_{A}}\\*{n^{\,2}_{\\!f}}\\*\bigg{(}{p_{gq}(x)}\\*\Big{(}-25\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,3}}-20\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,-2,0}}+25\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,2}}+8\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,0}}+26\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-2\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0,0}}-14\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0,0}}+27\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,0}}+16\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1,1}}+21\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}+4\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}-33\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\Big{)}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{p_{gq}(-x)}\\*\Big{(}-23\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,3}}+12\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-2,0}}+46\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,2}}-2\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2,0}}-20\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2,1}}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-58\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,-1,0}}+24\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,0,0}}+2\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0,0,0}}-75\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1}}+19\,\\*{\zeta_{2}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0}}+75\,\\*{\zeta_{3}}\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-1}}\Big{)}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{4}}\,\\*(36-16\,\\*x)+80\,\\*{\color[rgb]{1,0,0.87}{\rm{H}}_{-3,0}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,2}}\,\\*(192-48\,\\*x^{-1}-72\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,2}}\,\\*(-28+16\,\\*x^{-1}-2\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,0}}\,\\*(24+2\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3,1}}\,\\*(36+22\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,-1,0}}\,\\*(-192+48\,\\*x^{-1}+72\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0,0}}\,\\*(112$
$\displaystyle\mbox{{\hskip
8.53581pt}}-32\,\\*x^{-1}-48\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0,0}}\,\\*(112+32\,\\*x^{-1}+44\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,0}}\,\\*(-12+24\,\\*x^{-1}+22\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1,1}}\,\\*(-140+52\,\\*x^{-1}+4\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0,0}}\,\\*(12-6\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{3}}\,\\*(-1499/6-343/2\,\\*x+4/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2,0}}\,\\*(-956/3+212/3\,\\*x^{-1}+152\,\\*x-32/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,2}}\,\\*(-140/3-638/3\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+242/3\,\\*x-16\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,2}}\,\\*(277/3-311/3\,\\*x^{-1}-188/3\,\\*x+4/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,0}}\,\\*(67/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}-212/3\,\\*x^{-1}-265/6\,\\*x-4/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{2,1}}\,\\*(525/2-76\,\\*x^{-1}-7/4\,\\*x-12\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,-1,0}}\,\\*(460/3+958/3\,\\*x^{-1}-34/3\,\\*x+16\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0,0}}\,\\*(-260/3-550/3\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+50/3\,\\*x-32/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0,0}}\,\\*(-332/3-19/12\,\\*x+8/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0,0}}\,\\*(-184/3$
$\displaystyle\mbox{{\hskip
8.53581pt}}+434/3\,\\*x^{-1}-106/3\,\\*x-40/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,0}}\,\\*(283/3-83\,\\*x^{-1}-248/3\,\\*x-4/3\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1,1}}\,\\*(700/3-259\,\\*x^{-1}-299/3\,\\*x+44/3\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{-2}}\,\\*(72\,\\*{\zeta_{2}}\,\\*x^{-1}-288\,\\*{\zeta_{2}}+108\,\\*{\zeta_{2}}\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{2}}\,\\*(16495/36+1067/6\,\\*x^{-1}+610/9\,\\*x+1202/9\,\\*x^{2}-40\,\\*{\zeta_{2}}\,\\*x^{-1}-68\,\\*{\zeta_{2}}-34\,\\*{\zeta_{2}}\,\\*x)$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1,0}}\,\\*(226/3+4/3\,\\*x^{-2}+1498/9\,\\*x^{-1}+7/3\,\\*x+232/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0,0}}\,\\*(8735/18$
$\displaystyle\mbox{{\hskip
8.53581pt}}+3091/36\,\\*x+440/9\,\\*x^{2}-36\,\\*{\zeta_{2}}+16\,\\*{\zeta_{2}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,0}}\,\\*(1363/18-305/6\,\\*x^{-1}+301/18\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+52/9\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{1,1}}\,\\*(-8747/36+12383/36\,\\*x^{-1}+1498/9\,\\*x-10\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+{\color[rgb]{1,0,0.87}{\rm{H}}_{-1}}\,\\*(1117/3\,\\*{\zeta_{2}}\,\\*x^{-1}+370/3\,\\*{\zeta_{2}}-259/3\,\\*{\zeta_{2}}\,\\*x+24\,\\*{\zeta_{2}}\,\\*x^{2})+{\color[rgb]{1,0,0.87}{\rm{H}}_{0}}\,\\*(-10607/12$
$\displaystyle\mbox{{\hskip
8.53581pt}}-4916/27\,\\*x^{-1}-12139/72\,\\*x-9944/27\,\\*x^{2}+212/3\,\\*{\zeta_{2}}\,\\*x^{-1}+1499/6\,\\*{\zeta_{2}}+647/2\,\\*{\zeta_{2}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}-4/3\,\\*{\zeta_{2}}\,\\*x^{2}-72\,\\*{\zeta_{3}}\,\\*x^{-1}-72\,\\*{\zeta_{3}}-146\,\\*{\zeta_{3}}\,\\*x)+{\color[rgb]{1,0,0.87}{\rm{H}}_{1}}\,\\*(-4495/6+132025/216\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}+3983/36\,\\*x-6500/27\,\\*x^{2}-56\,\\*{\zeta_{2}}\,\\*x^{-1}-47/3\,\\*{\zeta_{2}}+205/3\,\\*{\zeta_{2}}\,\\*x+20/3\,\\*{\zeta_{2}}\,\\*x^{2})$
$\displaystyle\mbox{{\hskip
8.53581pt}}+30829/108-191575/648\,\\*x^{-1}-91561/432\,\\*x+27377/81\,\\*x^{2}-205/18\,\\*{\zeta_{2}}\,\\*x^{-1}$
$\displaystyle\mbox{{\hskip
8.53581pt}}-16495/36\,\\*{\zeta_{2}}-589/9\,\\*{\zeta_{2}}\,\\*x-1202/9\,\\*{\zeta_{2}}\,\\*x^{2}+104\,\\*{\zeta_{3}}\,\\*x^{-1}-752\,\\*{\zeta_{3}}+8587/12\,\\*{\zeta_{3}}\,\\*x$
$\displaystyle\mbox{{\hskip
8.53581pt}}+8\,\\*{\zeta_{3}}\,\\*x^{2}-114\,\\*{\zeta_{4}}\,\\*x^{-1}+331\,\\*{\zeta_{4}}-59/2\,\\*{\zeta_{4}}\,\\*x\bigg{)}\;,$

where we have suppressed the argument $x$ of the harmonic polylogarithms
(HPLs) $\,H_{m_{1},...,m_{w}}(x)$ and used the leading-order function
$\,{p_{gq}(x)}=2\,x^{\,-1}-2+x\,$ to shorten the $w=4$ part of the
expressions. For chains of indices zero we employ the abbreviated notation

$H_{{\>\\!\footnotesize\underbrace{0,\ldots,0}_{\scriptstyle m}},\,\pm
1,\,{\footnotesize\underbrace{0,\ldots,0}_{\scriptstyle n}},\,\pm
1,\,\ldots}(x)\;=\;H_{\pm(m+1),\,\pm(n+1),\,\ldots}(x)\>\>.$ (7)

Corresponding to the maximal weights in the $N$-space expression (4), i.e.,
$w=6$ for the ${C^{\,2}_{F}}\\*\,{n^{\,2}_{\\!f}}$ part and $w=5$ for the
$C_{F}C_{A}\\*\,{n^{\,2}_{\\!f}}$ contribution, the respective $x$-space
results in eq. (6) include HPLs up to $w=5$ and $w=4$. The term corresponding
to $\,(N\\!-\\!2)^{-1}\,S_{-2}(N\\!-\\!2)\,$ is $\,x^{\,-2}\,{\rm
H}_{-1,0}(x)$.

Of particular interest are the logarithmically enhanced endpoint contributions
in the high-energy (small-$x$) and threshold (large-$x$) limits. In the former
limit, the flavour-singlet splitting functions are dominated by the BFKL
single-log enhancement of the $x^{\,-1}$ terms. The subdominant $x^{\,0}$
contributions show a double-log enhancement. In both cases the leading
${n^{\,2}_{\\!f}}$ terms are of overall next-to-next-to-leading logarithmic
(NNLL) accuracy. Eq. (6) leads to

$\displaystyle P_{\rm gq}^{\,(3)}(x)\bigg{|}_{{n^{\,2}_{\\!f}}}$
$\displaystyle\\!\\!=\\!\\!$ $\displaystyle\frac{1}{x}\,\\*\ln
x\,\,\\*\bigg{[}{C^{\,2}_{F}}\\*\,{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{168448}{729}-\frac{6784}{81}\,\\*{\zeta_{2}}+\frac{512}{9}\,\\*{\zeta_{3}}\Big{)}$
(8) $\displaystyle\mbox{{\hskip 8.53581pt}{\hskip
8.53581pt}}\,+\,{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}-\frac{78080}{729}+\frac{3392}{81}\,\\*{\zeta_{2}}-\frac{128}{3}\,\\*{\zeta_{3}}\Big{)}\bigg{]}$
$\displaystyle\mbox{{}}+\frac{1}{x}\,\\*\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{1475572}{2187}+\frac{128}{3}\,\\*{\zeta_{2}}-\frac{1856}{81}\,\\*{\zeta_{3}}+\frac{2368}{27}\,\\*{\zeta_{4}}\Big{)}$
$\displaystyle\mbox{{\hskip 8.53581pt}{\hskip
8.53581pt}}\,+\,{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}-\frac{384878}{2187}-\frac{1640}{243}\,\\*{\zeta_{2}}+\frac{1664}{27}\,\\*{\zeta_{3}}-\frac{608}{9}\,\\*{\zeta_{4}}\Big{)}\bigg{]}$
$\displaystyle\mbox{{}}+\ln^{\,4\\!}x\,\,\\*\bigg{[}-\frac{1532}{81}\,\\*{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}+\frac{8}{27}\,\\*{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\bigg{]}$
$\displaystyle\mbox{{}}+\ln^{\,3\\!}x\,\,\\*\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{4120}{243}-\frac{832}{27}\,\\*{\zeta_{2}}\Big{)}\,-\frac{2848}{243}\\*{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\bigg{]}\;+\;\ldots$

where we have suppressed further terms for brevity. The $\ln^{\,4\\!}x$
contribution, which arise from the ${D_{0}^{\,5}}$ terms in eq. (4), agrees
with the prediction in eq. (5.11) of ref. [31], the rest is new.

The off-diagonal splitting functions exhibit a double logarithmic enhancement
also at large-$x$. In the present case it reads

$\displaystyle P_{\rm gq}^{\,(3)}(x)\bigg{|}_{{n^{\,2}_{\\!f}}}$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\ln^{\,4}(1\\!-\\!x)\,\\*\,\frac{32}{81}\,\\*{C_{F}}\,\\*{n^{\,2}_{\\!f}}\,\\*({C_{A}}-{C_{F}})\,-\,\ln^{\,3}(1\\!-\\!x)\,\\*\bigg{[}\,\frac{2404}{243}\,\\*{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}-\frac{2656}{243}\,\\*{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\bigg{]}$
(9)
$\displaystyle\mbox{{}{}}+\ln^{\,2}(1\\!-\\!x)\,\\*\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}-\frac{8870}{243}-\frac{32}{3}\,\\*{\zeta_{2}}\Big{)}+{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{18536}{243}+\frac{16}{27}\,\\*{\zeta_{2}}\Big{)}\bigg{]}$
$\displaystyle\mbox{{}{}}+\ln(1\\!-\\!x)\,\\*\,\bigg{[}\,{C^{\,2}_{F}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{1870}{81}-\frac{4144}{81}\,\\*{\zeta_{2}}-\frac{320}{27}\,\\*{\zeta_{3}}\Big{)}$
$\displaystyle\mbox{{\hskip 8.53581pt}{\hskip 8.53581pt}{\hskip
8.53581pt}{\hskip 8.53581pt}{\hskip
8.53581pt}}+{C_{F}}\,\\*{C_{A}}\,\\*{n^{\,2}_{\\!f}}\\*\Big{(}\,\frac{12866}{81}-\frac{160}{81}\,\\*{\zeta_{2}}+\frac{880}{27}\,\\*{\zeta_{3}}\Big{)}\bigg{]}\;+\;\ldots$

The $\ln^{\,4}(1\\!-\\!x)$ contribution, including the terms suppressed by
powers of ${(1\\!-\\!x)}$ not shown here, agrees with the predictions in eqs.
(5.16) and (5.25) of ref. [32], see also refs. [33, 34]. The remaining terms
in eq. (9) were not known before.

To summarize, we have derived the ${n^{\,2}_{\\!f}}$ contributions to
$P_{\,\rm gq}^{\,(3)}(x)$. Together with ref. [2] this completes the
${n^{\,2}_{\\!f}}$ parts of the 4-loop (N3LO) quark-to-parton splitting
functions for the evolution of unpolarized parton distributions of hadrons. As
usual, our results refer to the $\overline{\mbox{MS}}$ scheme, written in
terms of the expansion parameter $a_{\rm s}\,=\,\alpha_{\rm
s}({\mu^{\,2}})/(4\pi)$, i.e., we have, without loss or information,
identified the renormalization scale $\mu_{\>\\!\sf r}$ with the factorization
scale $\mu$ in eq. (9).

While the ${n^{\,2}_{\\!f}}$ parts are much larger, at any relevant number
${n_{\\!f}}$ of light flavours, than the corresponding leading
large-${n_{\\!f}}$ terms – see, e.g., the values at $N=8$ in eqs. (17) - 20 of
ref. [8] – these results are not yet of direct relevance for phenomenological
analyses of hard scattering processes.

It is not feasible to extend the present results to the complete N3LO
splitting functions with the method applied here. We expect it to be a
formidable task also for other approaches based on the method of differential
equations [35, 36, 37, 5, 2, 38]. Hence phenomenological analyses will have to
rely on approximate $x$-space expressions for the time being, as presented in
ref. [3] for the large-${n_{c}}$ suppressed non-singlet contributions and in
refs. [6, 7, 9] for the flavour-singlet cases. We hope to be able to present
results based on the even moments $2\leq N\leq 20$ also for the complete
$P_{\,\rm gq}^{\,(3)}(x)$ and $P_{\,\rm gg}^{\,(3)}(x)$ in the near future.

The biggest issue in such $x$-space approximations, as also noted in ref. [2],
are the hitherto unknown $x^{\,-1}\ln x$ contributions, i.e., the NNLL
corrections in the high-energy BFKL limit. With ref. [2] and the present
results, the first two such contributions have been derived. These results
should not only prove useful in the context of approximate expressions, but
also as checks of future calculations of NNLL corrections in the BFKL limit
and their transformation to the $\overline{\mbox{MS}}$ scheme, which in itself
will be a non-trivial operation, see refs. [39, 40]. Also the results for the
double-logarithmically enhanced contributions in the small-$x$ and large-$x$
limits can provide input for the extension of the respective resummations to a
higher logarithmic accuracy.

Form files with our results have been deposited at the preprint server
https://arXiv.org with the sources of this letter. They are also available
from the authors upon request.

## Acknowledgements

G.F. would like to acknowledge the influence of Marcello Ciafaloni, who passed
away recently, on his decision to work on quantum field theory. This work has
been supported by the Vidi grant 680-47-551 of the Dutch Research Council
(NWO), the UKRI FLF Mr/S03479x/1; the Consolidated Grant ST/P0000630/1
Particle Physics at the Higgs Centre of the UK Science and Technology
Facilities Council (STFC); the ERC Starting Grant 715049 QCDforfuture, the
EU’s Horizon Europe research and innovation programme under the Marie
Sklodowska-Curie grant 101104792, QCDchallenge; the Deutsche
Forschungsgemeinschaft through the Research Unit FOR 2926, Next Generation
pQCD for Hadron Structure: Preparing for the EIC, project number 40824754, and
DFG grant MO 1801/4-2, the ERC Advanced Grant 101095857 Conformal-EIC; and the
STFC Consolidated Grant ST/T000988/1. G.F. and S.M. are grateful to the
Galileo Galilei Institute in Florence for hospitality and support during the
scientific program on Theory Challenges in the Precision Era of the Large
Hadron Collider, where part of this work was done.

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