# Asymptotic normality of degree counts in a general preferential attachment
model

Simone Baldassarri Gianmarco Bet

###### Abstract

We consider the preferential attachment model. This is a growing random graph
such that at each step a new vertex is added and forms $m$ connections. The
neighbors of the new vertex are chosen at random with probability proportional
to their degree. It is well known that the proportion of nodes with a given
degree at step $n$ converges to a constant as $n\rightarrow\infty$. The goal
of this paper is to investigate the asymptotic distribution of the
fluctuations around this limiting value. We prove a central limit theorem for
the joint distribution of all degree counts. In particular, we give an
explicit expression for the asymptotic covariance. This expression is rather
complex, so we compute it numerically for various parameter choices. We also
use numerical simulations to argue that the convergence is quite fast. The
proof relies on the careful construction of an appropriate martingale.

##  1 Introduction

The unprecedented growth in size and complexity of social and economic
networks in the last two decades has sparked considerable interest in the
understanding of the fundamental properties of such networks. In this context,
the preferential attachment (PA) model, introduced in [4], is a well-known
model for a network that grows in time. More precisely, the PA model consists
of a sequence of graphs of increasing size such that each graph is obtained
from the previous according to a certain probabilistic rule. Namely, at each
step a new vertex is added and it forms connections with the vertices in the
graph in such a way that connections with vertices having larger degrees are
more likely. In fact, several different PA models appear in the literature,
depending on the concrete details of the attachment mechanism. For instance,
in [5, 9, 15, 19, 20, 22, 23, 25] the authors investigate a directed PA model,
while in [6, 8, 9, 14] an undirected version is considered. However, classical
PA networks do not always fit real-world network data well, or in many
applications it is natural to assign some kind of features to the vertices or
to the edges. This led to consider some extensions of the classical PA model.
For example, in [3] the authors consider a general family of preferential
attachment models with multi-type edges and in [1] with edge-steps, while [2,
11, 12, 24] investigate a PA model which mixes PA rules with uniform
attachment rules.

Here, we consider the PA model without self-loops described for example in
[8]. In particular, each graph in the sequence is connected. We make this
choice in order to simplify the calculations, but we believe that our result
also holds, for example, for the PA model with self-loops considered in the
standard reference [16, Chapter 8]. We argue this in more detail later on. We
assume that each new node has a fixed number $m\in\mathbb{N}$ of edges
attached to it. This is a particular case of the model considered in [8],
where $m$ is a random variable and is sampled for each new node. Our main
result is a central limit theorem for the proportion of nodes with a given
degree. In fact, we prove this jointly for all degree counts. In particular,
we give an explicit expression for the asymptotic covariance. Our results
generalize those obtained in [14] for the preferential attachment tree. More
precisely, in [14] the authors consider a PA model with self-loops and such
that $m=1$. Note that here we consider the PA model with $m\geq 1$ and without
self-loops. However, this does not influence the asymptotic behavior of the
degree counts, since as the graph size goes to infinity, the probability that
a new vertex forms a self-loop tends to zero. Because of this one would expect
to recover the results in [14] when plugging $m=1$ in our result. Indeed this
is the case if one takes into account a few minor mistakes in [14] which we
will discuss later. Note that a major difference between the two models is the
resulting connectivity structure. Our model produces a connected graph with
probability $1$ (w.p.$1$), while the model in [14] is disconnected w.p.$1$.
However, this does not play a role in the distribution of the degree counts.

In practice, not all nodes that enter the network have the same degree, and
thus it would be interesting to extend our result to the case of a random
initial degree distribution. Promising results on this model have been
obtained in [7, 8]. Moreover, in this paper we assume that the parameters of
the model are known, but in many practical situations one is given a
realization of the graph and the task is estimating the unknown parameters,
see [10, 17, 18, 21].

The outline of the paper is as follows: in Section 2 we define the model
rigorously and we state our main result. In Section 3 we give the proof of our
main result.

##  2 Model and results

Let us now describe in detail the random graph model that we consider. Fix
once and for all an integer $m\geq 1$. Formally, the preferential attachment
model is a sequence of random graphs $(\hbox{PA}_{s})_{s=1}^{t}$. The index
$s$ is interpreted as a time parameter. At time $s$, the graph $\hbox{PA}_{s}$
has a set $V=\\{0,1,...,s\\}$ of $s+1$ vertices. For $s=1$ the graph PA1
consists of the vertices 0 and 1, connected by $m$ edges. For $s\geq 2$, the
graph PAs is obtained from PAs-1 by adding a new vertex $s$ with degree $m$ as
follows. Define PA${}_{s,0}=$PAs-1 and PAs,1,…,PAs,m as the intermediate
graphs obtained by adding a new edge sequentially to PAs,0. For $i=1,...,m$,
PAs,i is obtained from PAs,i-1 by drawing an edge from $s$ to a randomly
selected vertex among $\\{0,1,...,s-1\\}$. The probability that a vertex $s$
is connected to some vertex $i$ is proportional to the degree of $i$. In other
words, vertices with large degrees are more likely to attract new edges. We
denote by $N_{k}(s,i)$ for $i=1,...,m$ the number of vertices of degree $k$
after the $i$-th edge has been attached at time $s$, excluding the vertex $s$.
We set $N_{k}(s+1,0):=N_{k}(s,m)$. Furthermore, we denote by $D_{s,i}$ the
degree of the vertex which has been attached to the $i-$th edge added when
constructing PAs from PAs-1. Consider now the $\sigma$-algebra ${\cal
F}_{s,i}$ generated by the preferential attachment construction up until the
attachment of the $i$-th edge of the new vertex at time $s$. The conditional
probability that the $i$-th edge connects to a vertex of degree $D_{s,i}$ is

${\mathbb{P}}(D_{s,i}=k|{\cal
F}_{s,i-1})=\frac{(k+{\delta})N_{k}(s,i-1)}{\sum_{j}(j+{\delta})N_{j}(s,i-1)},$
(2.1)

where ${\delta}>-m$ is an affine parameter. The normalizing constant in (2.1)
takes the simple form [8]

$\sum_{j=1}^{\infty}(j+{\delta})N_{j}(s,i-1)=s(2m+{\delta})-2m+i-1.$ (2.2)

For the standard PA model considered in [16, Chapter 8] it is shown that there
exists a probability mass function $\\{p_{k},\ k\geq m\\}$ such that,
uniformly on $i\in{\\{0,...,m\\}}$,

$\lim_{s\rightarrow\infty}\frac{N_{k}(s,i)}{s}=p_{k}\in{(0,1)},$ (2.3)

almost surely, where $p_{k}$ is given by

$p_{k}=(2+{\delta}/m)\frac{{\Gamma}(k+{\delta}){\Gamma}(m+2+{\delta}+{\delta}/m)}{{\Gamma}(m+{\delta}){\Gamma}(k+3+{\delta}+{\delta}/m)}.$
(2.4)

Here ${\Gamma}(\cdot)$ is the Gamma function. When the graph size goes to
infinity, the probability that a new vertex forms a self-loop tends to zero
and thus it easy to check that (2.3) and (2.4) hold still for our model
without self-loops following the proof proposed in [16, Section 8.6].

In order to state our main result, we require some further notation. We say
that the events $(A_{n})_{n}$ hold with high probability when
$P(A_{n})\rightarrow 1$ as $n\rightarrow\infty$. Given a random vector
$(X_{1}^{(n)},X_{2}^{(n)},...)$, we write
$(X_{1}^{(n)},X_{2}^{(n)},...)\Rightarrow(X_{1},X_{2},...)$ to indicate that
for any $k\in{\mathbb{N}}$, as $n\rightarrow\infty$
$(X_{1}^{(n)},X_{2}^{(n)},...,X_{k}^{(n)})$ converges to
$(X_{1},X_{2},...,X_{k})$ in distribution as vectors in ${\mathbb{R}}^{n}$.

Our main result is the following theorem.

max width=[mode=image—tex]figura1

Figure 1: $R_{Z}(r,r)$ for $m=1$ and ${\delta}=-0.5$, $0$, $1$, $3$, $5$ on
the left-hand side and $R_{Z}(r,r)$ for $m=2$ and ${\delta}=-1$, $0$, $2$,
$6$, $10$ on the right-hand side. To highlight the different behaviour of the
various variance functions, we use the logarithmic axis on the $y$-axis.

###### Theorem 2.5

As $s\rightarrow\infty$,

$\Bigg{(}\sqrt{s}\Big{(}\frac{N_{k}(s,i)}{s}-p_{k}\Big{)},\
k=m,m+1,...\Bigg{)}\Rightarrow(Z_{k},\ k=m,m+1,...),$ (2.6)

where $(Z_{k},\ k=m,m+1,...)$ is a mean zero Gaussian process with covariance
function $R_{Z}$ given by

$\begin{array}[]{lll}R_{Z}(r,l)\\\
=\frac{(-1)^{r+l}}{{\Gamma}(r+l+2m+1+3{\delta})}\displaystyle\sum_{q=m}^{\infty}(q+{\delta})p_{q}\Big{(}b_{m}^{(l)}b_{m}^{(r)}{\Gamma}(4m+3{\delta})(l+r-2m)!\\\
\quad\quad+(-1)^{m+q+1}(b_{m}^{(l)}b_{q}^{(r)}+b_{m}^{(r)}b_{q}^{(l)}){\Gamma}(q+3m+3{\delta})(l+r-q-m)!\\\
\quad\quad+(-1)^{m+q+1}(b_{m}^{(l)}b_{q+1}^{(r)}+b_{m}^{(r)}b_{q+1}^{(l)}){\Gamma}(q+3m+1+3{\delta})(l+r-m-q-1)!\\\
\quad\quad+mb_{q}^{(l)}b_{q}^{(r)}{\Gamma}(2q+2m+3{\delta})(r+l-2q)!\\\
\quad\quad+m(b_{q}^{(l)}b_{q+1}^{(r)}+b_{q}^{(r)}b_{q+1}^{(l)}){\Gamma}(2q+2m+1+3{\delta})(l+r-2q-1)!\\\
\quad\quad+mb_{q+1}^{(l)}b_{q+1}^{(r)}{\Gamma}(2q+2m+2+3{\delta})(l+r-2q-2)!\Big{)}\\\
\quad\quad-\frac{(2m+{\delta}){\Gamma}(l+{\delta}){\Gamma}(r+{\delta})}{({\Gamma}(m+{\delta}))^{2}}\displaystyle\sum_{t_{1}=m}^{l}\displaystyle\sum_{t_{2}=m}^{r}(-1)^{t_{1}+t_{2}}\\\
\quad\quad\times\frac{(t_{1}+t_{2}+2m+3{\delta})^{-1}(t_{1}+2+{\delta}-t_{1}/m)(t_{2}+2+{\delta}-t_{2}/m)}{(t_{1}-m)!(t_{2}-m)!(l-t_{1})!(r-t_{2})!(t_{1}+2+{\delta}+{\delta}/m)(t_{2}+2+{\delta}+{\delta}/m)}\\\
\quad\quad-\frac{(2m+{\delta})(m-1){\Gamma}(l+{\delta}){\Gamma}(r+{\delta})}{m^{2}({\Gamma}(m+{\delta}))^{2}}\displaystyle\sum_{t_{1}=m}^{l}\displaystyle\sum_{t_{2}=m}^{r}(-1)^{t_{1}+t_{2}}\\\
\quad\quad\times\frac{({\delta}+t_{1})({\delta}+t_{2})(t_{1}+t_{2}+2m+3{\delta})^{-1}}{(t_{1}-m)!(t_{2}-m)!(l-t_{1})!(r-t_{2})!(t_{1}+2+{\delta}+{\delta}/m)(t_{2}+2+{\delta}+{\delta}/m)},\\\
\end{array}$ (2.7)

for $r,l\geq m$. Here $b_{j}^{(k)}$ are given by

$b_{j}^{(k)}=\prod_{t=j}^{k-1}\frac{t+{\delta}}{t-k}=(-1)^{k-j}\frac{{\Gamma}(k+{\delta})}{(k-j)!{\Gamma}(j+{\delta})},\quad
1\leq j\leq k.$ (2.8)

When $m=1$, (2.7) reduces to [14, (4.28)], but their covariance functions
contains a few minor mistakes. More precisely, in the double sum the term
$(2+{\delta})^{2}$ should be $(2+{\delta})^{3}$ and the terms
${\Gamma}(l_{1}+2+2{\delta})$ and ${\Gamma}(l_{1}+2+2{\delta})$ should be
$(l_{1}+2+2{\delta})$ and $(l_{1}+2+2{\delta})$ respectively.

Since the expression in (2.7) is remarkably complicated, we plot it for
various parameter choices to help the understanding. In Figure 1 we plot the
function $r\mapsto R_{Z}(r,r)$ for fixed $m=1$ (resp. $m=2$) and various
values of ${\delta}$. On the other hand, in Figure 2 we plot the function
$R_{Z}(r,r)$ for fixed ${\delta}=0$ and various values of $m$. In Figure 3 we
plot the function $R_{Z}(r,5)$ for ${\delta}=0$ and $m=1$, $2$, $3$, $4$.
Finally, in Figure 4 on the left-hand- side we compare the asymptotic
covariance function $R_{Z}(r,r)$ for fixed $m=1,{\delta}=1$ with the empirical
variance obtained by numerically simulating the PA model up until time
$t=100,1000,5000$. Furthermore, in Figure 4 on the right-hand side, we compare
the asymptotic covariance function $R_{Z}(r,5)$ for fixed $m=2,{\delta}=0$
with the empirical covariance obtained by numerical simulation of the PA model
up until time $t=100,1000,5000$. While we do not have rigorous results on the
convergence speed of the rescaled vertex counts, Figure 4 suggests that the
convergence is indeed quite fast.

[mode=image—tex]figura2

Figure 2: $R_{Z}(r,r)$ for ${\delta}=0$ and $m=1,2,3$. To highlight the
different behaviour of the various variance functions, we use the logarithmic
axis on the $y$-axis.

max width=[mode=image—tex]figura3

Figure 3: $R_{Z}(r,5)$ for ${\delta}=0$ and $m=1$, $2$, $3$.

max width=[mode=image—tex]figura4

Figure 4: Left-hand side: $R_{Z}(r,r)$ for ${\delta}=1$ and $m=1$ ($t=\infty$)
compared with the numerical simulations stopped at time $t=100,1000,5000$.
Each empirical curve was obtained by taking the average of $N=10000$
simulations. To highlight the different behaviour of the various variance
functions, we use the logarithmic axis on the $y$-axis. Right-hand side:
$R_{Z}(r,5)$ for $m=2$ and ${\delta}=0$ compared with the numerical
simulations stopped at time $t=100,1000,5000$. Each empirical curve was
obtained by taking the average of $N=10000$ simulations.

##  3 Proof of the main result

In order to prove Theorem 2.5, we apply the following martingale central limit
theorem.

###### Theorem 3.1

[14] Let $\\{X_{n,m},{\cal F}_{m,n},1\leq m\leq n\\}$,
$X_{n,m}=(X_{n,m,1},...,X_{n,m,d})^{T}$ be a $d$-dimensional square-integrable
martingale difference array. Consider the $d\times d$ non-negative definite
random matrices

$G_{n,m}=({\mathbb{E}}[X_{n,m,i}X_{n,m,j}|{\cal F}_{n,m-1}],i,j=1,..,d),\quad
V_{n}=\displaystyle\sum_{m=1}^{n}G_{n,m},$ (3.2)

and suppose $(A_{n})$ is a sequence of $l\times d$ matrices with bounded
supremum norm. Assume that

  * (1)

$A_{n}V_{n}A_{n}^{T}\stackrel{{\scriptstyle P}}{{\rightarrow}}\Sigma$ as
$n\rightarrow\infty$ for some non-random (hence non-negative definite) matrix
$\Sigma$;

  * (2)

$\sum_{m\leq
n}{\mathbb{E}}[X^{2}{\mathds{1}}_{\\{{|X_{n,m,i}|>\varepsilon}\\}}|{\cal
F}_{n,m-1}]\stackrel{{\scriptstyle P}}{{\rightarrow}}0$ as
$n\rightarrow\infty$ for all $i=1,...,d$ and $\varepsilon>0$.

Then,

$\displaystyle\sum_{m=1}^{n}A_{n}X_{n,m}\Rightarrow X\quad as\
n\rightarrow\infty,$ (3.3)

in ${\mathbb{R}}^{l}$, where X is a mean zero l-dimensional Gaussian vector
with covariance matrix $\Sigma$.

We aim at constructing a martingale by appropriate rescaling of the process

$N_{k}(1),N_{k}(2,1),...,N_{k}(2,m-1),N_{k}(2,m),N_{k}(3,1),...,$

for all $k\geq m$. We have the recursion, for $s\geq 2$ and $0\leq i\leq m-1$,

$N_{k}(s,i+1)=N_{k}(s,i)-{\mathds{1}}_{\\{D_{s,i+1}=k\\}}+{\mathds{1}}_{\\{k=m\\}}{\mathds{1}}_{\\{i+1=m\\}}+{\mathds{1}}_{\\{k\neq
m\\}}{\mathds{1}}_{\\{D_{s,i+1=k-1}\\}}.$ (3.4)

The second term on the right-hand side (r.h.s) of (3.4) accounts for the
possibility that the edge being connected at time $i+1$ connects with some
vertex of degree $k$. The third term on the r.h.s of (3.4) takes into account
the new vertex joining at time $s$, after all of its $m$ edges have been
connected, while the fourth term accounts for the possibility that the edge
being connected at time $i+1$ might connect with some vertex of degree $k-1$.

For $s\geq 2$ and $0\leq i\leq m-1$, let us compute

$\begin{array}[]{lll}{\mathbb{E}}[N_{j}(s,i+1)|{\cal
F}_{i,s}]&=&N_{j}(s,i)-{\mathbb{P}}(D_{s,i+1}=j|{\cal
F}_{s,i})+{\mathds{1}}_{\\{j=m\\}}{\mathds{1}}_{\\{i+1=m\\}}\\\
&&+{\mathds{1}}_{\\{j\neq m\\}}{\mathbb{P}}(D_{s,i+1}=j-1|{\cal F}_{s,i})\\\
&=&N_{j}(s,i)-\frac{j+{\delta}}{s(2m+{\delta})-2m+i}N_{j}(s,i)\\\
&&+{\mathds{1}}_{\\{j=m\\}}{\mathds{1}}_{\\{i+1=m\\}}+{\mathds{1}}_{\\{j\neq
m\\}}\frac{j-1+{\delta}}{s(2m+{\delta})-2m+i}N_{j-1}(s,i)\\\
&=&N_{j}(s,i)\Big{(}1-\frac{j+{\delta}}{s(2m+{\delta})-2m+i}\Big{)}\\\
&&+{\mathds{1}}_{\\{j\neq
m\\}}\frac{j-1+{\delta}}{s(2m+{\delta})-2m+i}N_{j-1}(s,i)+{\mathds{1}}_{\\{j=m\\}}{\mathds{1}}_{\\{i+1=m\\}}.\end{array}$
(3.5)

We set

$M_{s,i}^{(k)}:=a_{s,i}^{(k)}\sum_{j=m}^{k}b_{j}^{(k)}(N_{j}(s,i)-{\mathbb{E}}[N_{j}(s,i)]),\quad
s\geq 2,\ 0\leq i\leq m.$ (3.6)

We will show that, for each $k\geq m$, the process $(M_{s,i}^{(k)})$ is a
martingale with respect to the same filtration. To this end, we compute

$\begin{array}[]{lll}{\mathbb{E}}[M_{s,i+1}^{(k)}|{\cal F}_{s,i}]\\\
={\mathbb{E}}\Bigg{[}a_{s,i+1}^{(k)}\displaystyle\sum_{j=m}^{k}b_{j}^{(k)}(N_{j}(s,i+1)-{\mathbb{E}}[N_{j}(s,i+1)])\Bigg{|}{\cal
F}_{s,i}\Bigg{]}\\\
=a_{s,i+1}^{(k)}b_{m}^{(k)}\Biggl{(}\Bigl{(}1-\frac{m+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{)}N_{m}(s,i)+{\mathds{1}}_{\\{i+1=m\\}}\\\
\quad-\Bigl{(}1-\frac{m+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{)}{\mathbb{E}}[N_{m}(s,i)]-{\mathds{1}}_{\\{i+1=m\\}}\Biggl{)}\\\
\quad+a_{s,i+1}^{(k)}\displaystyle\sum_{j=m+1}^{k}b_{j}^{(k)}\Biggl{(}N_{j}(s,i)\Bigl{(}1-\frac{j+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{)}\\\
\quad+\frac{j-1+{\delta}}{s(2m+{\delta})-2m+1}N_{j-1}(s,i)+\Bigl{(}1-\frac{j+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{)}{\mathbb{E}}[N_{j}(s,i)]\\\
\quad-\frac{j-1+{\delta}}{s(2m+{\delta})-2m+i}{\mathbb{E}}[N_{j-1}(s,i)]\Biggl{)}\\\
=a_{s,i+1}^{(k)}\displaystyle\sum_{j=m}^{k}\Biggl{(}b_{j}^{(k)}\Bigl{(}1-\frac{j+{\delta}}{s(2m+{\delta})-2m+1}\Bigl{)}\\\
\quad+b_{j+1}^{(k)}\frac{j+{\delta}}{s(2m+{\delta})-2m+1}\Biggl{)}(N_{j}(s,i)-{\mathbb{E}}[N_{j}(s,i)])\\\
\quad+b_{k}^{(k)}\Bigl{(}1-\frac{k+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{)}(N_{k}(s,i)-{\mathbb{E}}[N_{k}(s,i)]).\end{array}$
(3.7)

This is the same as the right-hand side of (3.6) if the following holds:

$\left\\{\begin{array}[]{@{}l@{}}a_{s,i+1}^{(k)}\Bigl{[}b_{j}^{(k)}\Bigl{(}1-\frac{j+{\delta}}{s(2m+{\delta})-2m+1}\Bigl{)}+b_{j+1}^{(k)}\frac{j+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{]}=a_{s,i}^{(k)}b_{j}^{(k)},\
j=m,..,k-1\\\
a_{s,i+1}^{(k)}b_{k}^{(k)}\Bigl{(}1-\frac{k+{\delta}}{s(2m+{\delta})-2m+i}\Bigl{)}=a_{s,i}^{(k)}b_{k}^{(k)}.\end{array}\right.$
(3.8)

Let

$\begin{array}[]{lll}a_{s,i}^{(k)}&:=&\Biggl{[}\displaystyle\prod_{t=k-m+1}^{s-1}\prod_{r=0}^{m-1}\Biggl{(}1-\frac{k+{\delta}}{t(2m+{\delta})-2m+r}\Biggl{)}\Biggl{]}^{-1}\\\
&&\times\Biggl{[}\displaystyle\prod_{r=0}^{i-1}\Biggl{(}1-\frac{k+{\delta}}{s(2m+{\delta})-2m+r}\Biggl{)}\Biggl{]}^{-1}.\end{array}$
(3.9)

Let also

$b_{j}^{(k)}:=\prod_{t=j}^{k-1}\frac{t+{\delta}}{t-k}=(-1)^{k-j}\frac{{\Gamma}(k+{\delta})}{(k-j)!{\Gamma}(j+{\delta})},\quad
1\leq j\leq k.$ (3.10)

We adopt the usual convention that $b_{k}^{(k)}=1$ and $b_{j}^{(k)}=0$ for all
$j>k$. Direct calculations show that (3.9) and (3.10) satisfy the conditions
in (3.8).

Note that

$a_{s,i+1}^{(k)}=a_{s,i}^{(k)}\Biggl{(}1-\frac{k+{\delta}}{s(2m+{\delta})-2m+i}\Biggl{)}^{-1}.$
(3.11)

Moreover, by the Stirling formula, as $s\rightarrow\infty$,

$\begin{array}[]{lll}a_{s,i}^{(k)}&=&\displaystyle\prod_{r=0}^{m-1}\frac{{\Gamma}\Big{(}s+\frac{r-2m}{2m+{\delta}}\Big{)}\Big{/}{\Gamma}\Big{(}k-m+1+\frac{r-2m}{2m+{\delta}}\Big{)}}{{\Gamma}\Big{(}s+\frac{r-k-{\delta}-2m}{2m+{\delta}}\Big{)}\Big{/}{\Gamma}\Big{(}k-m+1+\frac{r-k-{\delta}-2m}{2m+{\delta}}\Big{)}}\\\
&&\times\displaystyle\prod_{r=0}^{i-1}\Big{(}1-\frac{k+{\delta}}{s(2m+{\delta})-2m+r}\Big{)}^{-1}\\\
&\sim&s^{\frac{k+{\delta}}{2m+{\delta}}}\displaystyle\prod_{r=0}^{m-1}\frac{{\Gamma}\Big{(}k-m+1+\frac{r-k-{\delta}-2m}{2m+{\delta}}\Big{)}}{{\Gamma}\Big{(}k-m+1+\frac{r-2m}{2m+{\delta}}\Big{)}},\end{array}$
(3.12)

so that $s\to a_{s,i}^{(k)}$ is regularly varying with index
$(k+{\delta})/(2m+{\delta})$.

Thus we have proved that the process $(M_{s,i}^{(k)})$ is a martingale with
respect to the filtration $({\cal F}_{s,i-1})$.

###### Remark 3.13

Note that there is not a unique set of coefficients $a_{s,i}^{(k)}$ and
$b_{j}^{(k)}$ that satisfies (3.8). For example, if
$a_{s,i}^{(k)},b_{j}^{(k)}$ is a solution of (3.8), then also $\alpha
a_{s,i}^{(k)},b_{j}^{(k)}$ is a solution for any $\alpha\in{\mathbb{R}}$.
However, if we require that the boundary conditions for $M_{s,i}^{(k)}$ are
satisfied, i.e., $M_{s,m}^{(k)}=M_{s+1,0}^{(k)}$ for any $k\geq m$, then (3.8)
admits a unique solution. This implies that the indexing notations for the
processes $N_{j}(s,i)$ and $M_{s,i}^{(k)}$ are consistent.

There exists a probability mass function $\\{p_{k},\ k\geq m\\}$ such that as
$s\rightarrow\infty$ [16, Chapter 8]

$\frac{N_{k}(s,i)}{s}\rightarrow p_{k},\quad k\geq m,$ (3.14)

almost surely, uniformly over $i\in{\\{0,...,m-1\\}}$.

For $k=m,m+1,...$ and $i\in\\{0,...,m-1\\}$, define a $k$-variate triangular
array of martingale differences by

$X_{s,h,j}^{(l)}:=\frac{M_{h,j}^{(l)}-M_{h,j-1}^{(l)}}{a_{s,j}^{(l)}s^{1/2}},\quad
h=k+1,..,s;\ l=m,..,k;\ j=0,..,i.$ (3.15)

In order to use the multivariate martingale central limit theorem, we compute
the asymptotic form of the quantities

$\displaystyle G_{s,h,j}(r,l)$
$\displaystyle:={\mathbb{E}}[X_{s,h,j}^{(r)}X_{s,h,j}^{(l)}|{\cal F}_{h,i-1}]$
(3.16)
$\displaystyle=(a_{s,j}^{(r)}a_{s,j}^{(l)}s)^{-1}{\mathbb{E}}\Bigl{[}(M_{h,j}^{(r)}-M_{h,j-1}^{(r)})(M_{h,j}^{(l)}-M_{h,j-1}^{(l)})\Bigl{|}{\cal
F}_{h,j-1}\Bigl{]}$

By the martingale property we get

$\begin{array}[]{lll}{\mathbb{E}}\Bigl{[}(M_{h,j}^{(r)}-M_{h,j-1}^{(r)})(M_{h,j}^{(l)}-M_{h,j-1}^{(l)})\Bigl{|}{\cal
F}_{h,j-1}\Bigl{]}\\\
={\mathbb{E}}\Biggl{[}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}(a_{h,j}^{(r)}N_{d}(h,j))-a_{h,j-1}^{(r)}N_{d}(h,j-1))\\\
\quad\times\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(a_{h,j}N_{t}(h,j)-a_{h,j-1}N_{t}(h,j-1))\Biggl{|}{\cal
F}_{h,j-1}\Biggl{]}\\\
\quad-\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}(a_{h,j}^{(r)}{\mathbb{E}}[N_{d}(h,j)]-a_{h,j-1}^{(r)}{\mathbb{E}}[N_{d}(h,j-1)])\\\
\quad\times\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(a_{h,j}^{(l)}{\mathbb{E}}[N_{t}(h,j)]-a_{h,j-1}^{(l)}{\mathbb{E}}[N_{t}(h,j-1)]).\end{array}$
(3.17)

We begin by analyzing the behaviour of the deterministic term on the right-
hand side of (3.17). When $d=m$, by (3.11) we obtain

$\begin{array}[]{lll}a_{s,j}^{(l)}{\mathbb{E}}[N_{m}(s,j)]-a_{s,j-1}^{(l)}{\mathbb{E}}[N_{m}(s,j-1)]\\\
=a_{s,j}^{(l)}\Bigl{[}{\mathds{1}}_{\\{j=m\\}}+\Bigl{(}1-\frac{m+{\delta}}{s(2m+{\delta})-2m+j-1}\Bigl{)}{\mathbb{E}}[N_{m}(s,j-1)]\Bigl{]}-a_{s,j-1}^{(l)}{\mathbb{E}}[N_{m}(s,j-1)]\\\
=a_{s,j}^{(l)}\Bigl{[}{\mathds{1}}_{\\{j=m\\}}+{\mathbb{E}}[N_{m}(s,j-1)]\frac{l-m}{s(2m+{\delta})-2m+j-1}\Bigl{]}\\\
\sim
a_{s,j-1}^{(l)}\Bigl{(}{\mathds{1}}_{\\{j=m\\}}+p_{m}\frac{l-m}{2m+{\delta}}\Bigl{)},\end{array}$
(3.18)

almost surely as $s\rightarrow\infty$, since $a_{s,j}^{(l)}\sim
a_{s,j-1}^{(l)}$.

Now consider the case $d\geq m+1$:

$\begin{array}[]{lll}a_{s,j}^{(l)}{\mathbb{E}}[N_{d}(s,j)]-a_{s,j-1}^{(l)}{\mathbb{E}}[N_{d}(s,j-1)]\\\
=a_{s,j}^{(l)}\Bigl{(}\Bigl{(}1-\frac{d+{\delta}}{s(2m+{\delta})-2m+j-1}\Bigl{)}{\mathbb{E}}[N_{d}(s,j-1)]+\frac{d-1+{\delta}}{s(2m+{\delta})-2m+j-1}{\mathbb{E}}[N_{d-1}(s,j-1)]\Bigl{)}\\\
\quad-a_{s,j-1}^{(l)}{\mathbb{E}}[N_{d}(s,j-1)]\\\
=a_{s,j}^{(l)}\Bigl{(}\frac{d-1+{\delta}}{s(2m+{\delta})-2m+j-1}{\mathbb{E}}[N_{d-1}(s,j-1)]+\frac{l-d}{s(2m+{\delta})-2m+j-1}{\mathbb{E}}[N_{d}(s,j-1)]\Bigl{)}\\\
\sim\frac{a_{s,j-1}^{(l)}}{2m+{\delta}}\Big{(}(l-d)p_{d}+(d-1+{\delta})p_{d-1}\Big{)},\end{array}$
(3.19)

almost surely as $s\rightarrow\infty$. Recall that $p_{d}$ is given by [16,
(8.42)].

In conclusion, we get

$\begin{array}[]{lll}\displaystyle\lim_{s\rightarrow\infty}\frac{1}{a_{s,j-1}^{(l)}}\displaystyle\sum_{d=m}^{l}b_{d}^{(l)}\Bigl{(}a_{s,j}^{(l)}{\mathbb{E}}[N_{d}(s,j)]-a_{s,j-1}^{(l)}{\mathbb{E}}[N_{d}(s,j-1)]\Bigl{)}\\\
=b_{m}^{(l)}{\mathds{1}}_{\\{j=m\\}}+\frac{1}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(l)}[(l-d)p_{d}+(d-1+{\delta})p_{d-1}]\end{array}$
(3.20)

Let us focus on the second term of the right-hand side of (3.20). Rearranging
the terms in the sum,

$\begin{array}[]{lll}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}[(r-d)p_{d}+(d-1+{\delta})p_{d-1}]&=&\displaystyle\sum_{d=m}^{r-1}p_{d}[(r-d)b_{d}^{(r)}+(d+{\delta})b_{d+1}^{(r)}]\\\
&=&\displaystyle\sum_{d=m}^{r-1}p_{d}b_{d}^{(r)}\Bigg{[}r-d+(d+{\delta})\frac{b_{d+1}^{(r)}}{b_{d}^{(r)}}\Bigg{]}\\\
&=&0,\par\end{array}$ (3.21)

since by (3.10)

$\frac{b_{d+1}^{(r)}}{b_{d}^{(r)}}=\frac{d-r}{d+{\delta}}.$ (3.22)

Summarizing, we get

$\displaystyle\lim_{s\rightarrow\infty}\frac{1}{a_{s,j-1}^{(l)}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}\Bigl{(}a_{s,j}^{(r)}{\mathbb{E}}[N_{d}(s,j)]-a_{s,j-1}^{(r)}{\mathbb{E}}[N_{d}(s,j-1)]\Bigl{)}\\\
=b_{m}^{(r)}{\mathds{1}}_{\\{j=m\\}}.$ (3.23)

Next we look at the first term on the right-hand side of (3.17). By (3.4) we
obtain:

$\begin{array}[]{lll}a_{s,j}^{(r)}N_{d}(s,j)-a_{s,j-1}^{(r)}N_{d}(s,j-1)\\\
=a_{s,j}^{(r)}\Bigl{(}N_{d}(s,j-1)+{\mathds{1}}_{\\{d=m\\}}{\mathds{1}}_{\\{j=m\\}}+{\mathds{1}}_{\\{d\neq
m\\}}{\mathds{1}}_{\\{D_{s,j}=d-1\\}}-{\mathds{1}}_{\\{D_{s,j}=d\\}}\Bigl{)}\\\
\quad-a_{s,j-1}^{(r)}N_{d}(s,j-1)\\\
=a_{s,j}^{(r)}\Bigl{(}N_{d}(s,j-1)\frac{r+{\delta}}{s(2m+{\delta})-2m+j-1}-{\mathds{1}}_{\\{D_{s,j}=d\\}}+{\mathds{1}}_{\\{d=m\\}}{\mathds{1}}_{\\{j=m\\}}\\\
\quad+{\mathds{1}}_{\\{d\neq
m\\}}{\mathds{1}}_{\\{D_{s,j}=d-1\\}}\Bigl{)}:=a_{s,j}^{(r)}\Bigl{(}N_{d}(s,j-1)\frac{r+{\delta}}{s(2m+{\delta})-2m+j-1}+B_{s}^{(j)}(d)\Bigl{)},\end{array}$
(3.24)

where

$B_{s}^{(j)}(d):=-{\mathds{1}}_{\\{D_{s,j}=d\\}}+{\mathds{1}}_{\\{d=m\\}}{\mathds{1}}_{\\{j=m\\}}+{\mathds{1}}_{\\{d\neq
m\\}}{\mathds{1}}_{\\{D_{s,j}=d-1\\}}.$ (3.25)

Thus we get

$\begin{array}[]{lll}{\mathbb{E}}\Bigg{[}\frac{1}{a_{s,j}^{(r)}a_{s,j}^{(l)}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}(a_{s,j}^{(r)}N_{d}(s,j)-a_{s,j-1}^{(r)}N_{d}(s,j-1))\\\
\quad\times\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(a_{s,j}^{(l)}N_{t}(s,j)-a_{s,j-1}^{(l)}N_{t}(s,j-1))\Bigg{|}{\cal
F}_{s,j-1}\Bigg{]}\\\
={\mathbb{E}}\Bigg{[}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}\Bigg{(}N_{d}(s,j-1)\frac{r+{\delta}}{s(2m+{\delta})-2m+j-1}+B_{s}^{(j)}(d)\Bigg{)}\times\\\
\quad\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}\Bigg{(}N_{t}(s,j-1)\frac{l+{\delta}}{s(2m+{\delta})-2m+j-1}+B_{s}^{(j)}(t)\Bigg{)}\Bigg{|}{\cal
F}_{s,j-1}\Bigg{]}\end{array}$ (3.26)

By direct calculations, the r.h.s of (3.26) becomes

$\begin{array}[]{lll}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}(r+{\delta})\frac{N_{d}(s,j-1)}{s(2m+{\delta})-2m+j-1}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(l+{\delta})\frac{N_{t}(s,j-1)}{s(2m+{\delta})-2m+j-1}\\\
\quad+\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}(r+{\delta})\frac{N_{d}(s,j-1)}{s(2m+{\delta})-2m+j-1}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}{\mathbb{E}}[B_{s}^{(j)}(t)|{\cal
F}_{s,j-1}]\\\
\quad+\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(l+{\delta})\frac{N_{t}(s,j-1)}{s(2m+{\delta})-2m+j-1}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}{\mathbb{E}}[B_{s}^{(j)}(d)|{\cal
F}_{s,j-1}]\\\
\quad+\displaystyle\sum_{d=m}^{r}\displaystyle\sum_{t=m}^{l}b_{d}^{(r)}b_{t}^{(l)}{\mathbb{E}}[B_{s}^{(j)}(d)B_{s}^{(j)}(t)|{\cal
F}_{s,j-1}]\\\
:=S_{1,s}^{(j)}(r,l)+S_{2,s}^{(j)}(r,l)+S_{3,s}^{(j)}(r,l)+S_{4,s}^{(j)}(r,l),\end{array}$
(3.27)

We are interested in the behaviour of the terms $S_{1,s}^{(j)}(r,l)$,
$S_{2,s}^{(j)}(r,l)$, $S_{3,s}^{(j)}(r,l)$ and $S_{4,s}^{(j)}(r,l)$ as
$s\rightarrow\infty$. We first deal with the terms $S_{1,s}^{(j)}(r,l)$,
$S_{2,s}^{(j)}(r,l)$, $S_{3,s}^{(j)}(r,l)$. To this end, note that with
probability 1 as $s\rightarrow\infty$

$\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}(r+{\delta})\frac{N_{d}(s,j-1)}{s(2m+{\delta})+j-1}\sim\frac{r+{\delta}}{2m+{\delta}}\sum_{d=m}^{r}b_{d}^{(r)}p_{d}$
(3.28)

and that

$\begin{array}[]{lll}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}{\mathbb{E}}[B_{s}^{(j)}(d)|{\cal
F}_{s,j-1}]\\\
=b_{m}^{(r)}{\mathbb{E}}[{\mathds{1}}_{\\{j=m\\}}-{\mathds{1}}_{\\{D_{s,j}=m\\}}|{\cal
F}_{s,j-1}]\\\
\quad+\displaystyle\sum_{d=m+1}^{r}b_{d}^{(r)}{\mathbb{E}}[{\mathds{1}}_{\\{D_{s,j}=d-1\\}}-{\mathds{1}}_{\\{D_{s,j}=d\\}}|{\cal
F}_{s,j-1}]\\\
=b_{m}^{(r)}{\mathds{1}}_{\\{j=m\\}}-b_{m}^{(r)}{\mathbb{P}}(D_{s,j}=m|{\cal
F}_{s,j-1})\\\
\quad+\displaystyle\sum_{d=m+1}^{r}b_{d}^{(r)}({\mathbb{P}}(D_{s,j}=d-1|{\cal
F}_{s,j-1})-{\mathbb{P}}(D_{s,j}=d|{\cal F}_{s,j-1}))\\\
=b_{m}^{(r)}\Big{(}{\mathds{1}}_{\\{j=m\\}}-\frac{m+{\delta}}{s(2m+{\delta})-2m+j-1}N_{m}(s,j-1)\Big{)}\\\
\quad+\displaystyle\sum_{d=m+1}^{r}b_{d}^{(r)}\Big{(}\frac{d-1+{\delta}}{s(2m+{\delta})-2m+j-1}N_{d-1}(s,j-1)\\\
\quad-\frac{d+{\delta}}{s(2m+{\delta})-2m+j-1}N_{d}(s,j-1)\Big{)}\\\
\end{array}$ (3.29)

Thus we obtain

$\begin{array}[]{lll}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}{\mathbb{E}}[B_{s}^{(j)}(d)|{\cal
F}_{s,j-1}]\\\ \sim
b_{m}^{(r)}{\mathds{1}}_{\\{j=m\\}}+\frac{1}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}((d-1+{\delta})p_{d-1}-(d+{\delta})p_{d})\\\
=b_{m}^{(r)}{\mathds{1}}_{\\{j=m\\}}-\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d},\end{array}$
(3.30)

almost surely as $s\rightarrow\infty$, where at the last step we used (3.21).
Summarizing, with high probability

$\begin{array}[]{lll}S_{1,s}^{(j)}(r,l)\rightarrow\frac{(r+{\delta})(l+{\delta})}{(2m+{\delta})^{2}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}^{,}\\\
S_{2,s}^{(j)}(r,l)\rightarrow\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\Bigg{(}b_{m}^{(l)}{\mathds{1}}_{\\{j=m\\}}-\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Bigg{)},\\\
S_{3,s}^{(j)}(r,l)\rightarrow\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Bigg{(}b_{m}^{(r)}{\mathds{1}}_{\\{j=m\\}}-\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\Bigg{)},\end{array}$
(3.31)

as $s\rightarrow\infty$. Finally, we consider the term $S_{4,s}(r,l)$. Note
that, by (3.25), we have

$\left\\{\begin{array}[]{@{}l@{}}B_{s}^{(j)}(m)={\mathds{1}}_{\\{j=m\\}}-{\mathds{1}}_{\\{D_{s,j}=m\\}},\\\
B_{s}^{(j)}(d)={\mathds{1}}_{\\{D_{s,j}=d-1\\}}-{\mathds{1}}_{\\{D_{s,j}=d\\}},\quad
d\geq m+1.\end{array}\right.$ (3.32)

We separate the analysis of these terms according to the following two events:

  1. a)

$\\{D_{s,j}=m\\}$,

  2. b)

$\\{D_{s,j}=h\\}$, $h\geq m+1$.

On the event $\\{D_{s,j}=m\\}$, by (3.25), we get

$\begin{array}[]{lll}{\mathbb{E}}[B_{s}^{(j)}(d)B_{s}^{(j)}(t)|{\cal
F}_{s,j-1}]&=&{\mathds{1}}_{\\{d=t=m\\}}{\mathds{1}}_{\\{j\neq
m\\}}+{\mathds{1}}_{\\{d=t=m+1\\}}-{\mathds{1}}_{\\{d=m\\}}\\\
&&\times{\mathds{1}}_{\\{t=m+1\\}}{\mathds{1}}_{\\{j\neq
m\\}}-{\mathds{1}}_{\\{t=m\\}}{\mathds{1}}_{\\{d=m+1\\}}{\mathds{1}}_{\\{j\neq
m\\}}.\end{array}$ (3.33)

On $\\{D_{s,j}=h\\}$, $h\geq m+1$ we get

$\begin{array}[]{lll}{\mathbb{E}}[B_{s}^{(j)}(d)B_{s}^{(j)}(t)|{\cal
F}_{s,j-1}]\\\
={\mathds{1}}_{\\{d=t=m\\}}{\mathds{1}}_{\\{j=m\\}}+{\mathds{1}}_{\\{d=m\\}}{\mathds{1}}_{\\{j=m\\}}\Big{(}{\mathds{1}}_{\\{t=h+1\\}}-{\mathds{1}}_{\\{t=h\\}}\Big{)}\\\
\quad+{\mathds{1}}_{\\{t=m\\}}{\mathds{1}}_{\\{j=m\\}}\Big{(}{\mathds{1}}_{\\{d=h+1\\}}-{\mathds{1}}_{\\{d=h\\}}\Big{)}+{\mathds{1}}_{\\{t=d=h\\}}+{\mathds{1}}_{\\{t=d=h+1\\}}\\\
\quad-{\mathds{1}}_{\\{d=h\\}}{\mathds{1}}_{\\{t=h+1\\}}-{\mathds{1}}_{\\{d=h+1\\}}{\mathds{1}}_{\\{t=h\\}}.\end{array}$
(3.34)

By (3.27), combining (3.33) and (3.34) we obtain

$\begin{array}[]{lll}S_{4,s}^{(j)}(r,l)&=&\frac{m+{\delta}}{s(2m+{\delta})-2m+j-1}N_{m}(s,j-1)\Big{[}{\mathds{1}}_{\\{j\neq
m\\}}(b_{m}^{(r)}b_{m}^{(l)}-b_{m}^{(r)}b_{m+1}^{(l)}\\\
&&-b_{m+1}^{(r)}b_{m}^{(l)})+b_{m+1}^{(r)}b_{m+1}^{(l)}\Big{]}+\displaystyle\sum_{h=m+1}^{s}\frac{h+{\delta}}{s(2m+{\delta})-2m+j-1}\\\
&&\times
N_{h}(s,j-1)\Big{[}{\mathds{1}}_{\\{j=m\\}}(b_{m}^{(r)}b_{m}^{(l)}+b_{m}^{(r)}(b_{h+1}^{(l)}-b_{h}^{(l)})\\\
&&+b_{m}^{(l)}(b_{h+1}^{(r)}-b_{h}^{(r)}))+b_{h}^{(r)}b_{h}^{(l)}+b_{h+1}^{(r)}b_{h+1}^{(l)}-b_{h}^{(r)}b_{h+1}^{(l)}-b_{h+1}^{(r)}b_{h}^{(l)}\Big{]}\\\
&\sim&\frac{m+{\delta}}{2m+{\delta}}p_{m}\Big{[}{\mathds{1}}_{\\{j\neq
m\\}}(b_{m}^{(r)}b_{m}^{(l)}-b_{m}^{(r)}b_{m+1}^{(l)}-b_{m+1}^{(r)}b_{m}^{(l)})+b_{m+1}^{(r)}b_{m+1}^{(l)}\Big{]}\\\
&&+\displaystyle\sum_{h=m+1}^{\infty}\frac{h+{\delta}}{2m+{\delta}}p_{h}\Big{[}{\mathds{1}}_{\\{j=m\\}}(b_{m}^{(r)}b_{m}^{(l)}+b_{m}^{(r)}(b_{h+1}^{(l)}-b_{h}^{(l)})\\\
&&+b_{m}^{(l)}(b_{h+1}^{(r)}-b_{h}^{(r)}))+b_{h}^{(r)}b_{h}^{(l)}+b_{h+1}^{(r)}b_{h+1}^{(l)}-b_{h}^{(r)}b_{h+1}^{(l)}-b_{h+1}^{(r)}b_{h}^{(l)}\Big{]},\end{array}$
(3.35)

almost surely as $s\rightarrow\infty$.

Replacing in (3.27) the expressions derived in (3.31) for the terms
$S_{1,s}^{(j)}(r,l)$, $S_{2,s}^{(j)}(r,l)$ and $S_{3,s}^{(j)}(r,l)$ and in
(3.35) for $S_{4,s}^{(j)}(r,l)$, we conclude that, with high probability,

$\begin{array}[]{lll}{\mathbb{E}}\Big{[}\frac{1}{a_{s,j}^{(r)}a_{s,j}^{(l)}}(M_{s,j}^{(r)}-M_{s,j-1}^{(r)})(M_{s,j}^{(l)}-M_{s,j-1}^{(l)})\Big{|}{\cal
F}_{s,j-1}\Big{]}\rightarrow a^{(j)}(r,l)\\\
:=\frac{m+{\delta}}{2m+{\delta}}p_{m}\Big{(}{\mathds{1}}_{\\{j\neq
m\\}}(b_{m}^{(r)}b_{m}^{(l)}-b_{m}^{(r)}b_{m+1}^{(l)}-b_{m+1}^{(r)}b_{m}^{(l)})+b_{m+1}^{(r)}b_{m+1}^{(l)}\Big{)}\\\
\quad+\displaystyle\sum_{h=m+1}^{\infty}\frac{h+{\delta}}{2m+{\delta}}p_{h}\Big{(}{\mathds{1}}_{\\{j=m\\}}(b_{m}^{(r)}b_{m}^{(l)}+b_{m}^{(r)}(b_{h+1}^{(l)}-b_{h}^{(l)})+b_{m}^{(l)}(b_{h+1}^{(r)}\\\
\quad-
b_{h}^{(r)}))+b_{h}^{(r)}b_{h}^{(l)}+b_{h+1}^{(r)}b_{h+1}^{(l)}-b_{h}^{(r)}b_{h+1}^{(l)}-b_{h+1}^{(r)}b_{h}^{(l)}\Big{)}-\Big{(}b_{m}^{(r)}{\mathds{1}}_{\\{j=m\\}}\\\
\quad-\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\Big{)}\Big{(}b_{m}^{(l)}{\mathds{1}}_{\\{j=m\\}}-\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Big{)}.\end{array}$
(3.36)

Now we consider the quantity

$\begin{array}[]{lll}a(r,l)&:=&\displaystyle\sum_{j=0}^{m-1}a^{(j)}(r,l)\\\
&=&\frac{m+{\delta}}{2m+{\delta}}p_{m}\Big{(}(m-1)(b_{m}^{(r)}b_{m}^{(l)}-b_{m}^{(r)}b_{m+1}^{(l)}-b_{m+1}^{(r)}b_{m}^{(l)})+mb_{m+1}^{(r)}b_{m+1}^{(l)}\Big{)}\\\
&&+\displaystyle\sum_{h=m+1}^{\infty}\frac{h+{\delta}}{2m+{\delta}}p_{h}\Big{(}b_{m}^{(r)}b_{m}^{(l)}+b_{m}^{(r)}(b_{h+1}^{(l)}-b_{h}^{(l)})\\\
&&+b_{m}^{(l)}(b_{h+1}^{(r)}-b_{h}^{(r)})+m(b_{h}^{(r)}b_{h}^{(l)}+b_{h+1}^{(r)}b_{h+1}^{(l)}-b_{h}^{(r)}b_{h+1}^{(l)}-b_{h+1}^{(r)}b_{h}^{(l)})\Big{)}\\\
&&-\Big{(}b_{m}^{(r)}-\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\Big{)}\Big{(}b_{m}^{(l)}-\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Big{)}\\\
&&-(m-1)\frac{(r+{\delta})(l+{\delta})}{(2m+{\delta})^{2}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\\\
&=&\displaystyle\sum_{h=m}^{\infty}\frac{h+{\delta}}{2m+{\delta}}p_{h}[(b_{m}^{(r)}+b_{h+1}^{(r)}-b_{h}^{(r)})(b_{m}^{(l)}+b_{h+1}^{(l)}-b_{h}^{(l)})\\\
&&+(m-1)(b_{h+1}^{(l)}-b_{h}^{(l)})(b_{h+1}^{(r)}-b_{h}^{(r)})]\\\
&&-\Big{(}b_{m}^{(r)}-\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\Big{)}\Big{(}b_{m}^{(l)}-\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Big{)}\\\
&&-(m-1)\frac{(r+{\delta})(l+{\delta})}{(2m+{\delta})^{2}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}.\end{array}$
(3.37)

By [16, Section 8], we can write

$\begin{array}[]{lll}p_{d}&=&\Bigg{(}(2+{\delta}/m)\frac{{\Gamma}(m+2+{\delta}+{\delta}/m)}{{\Gamma}(m+{\delta})}\Bigg{)}\frac{{\Gamma}(d+{\delta})}{{\Gamma}(d+3+{\delta}+{\delta}/m)}\\\
&:=&c({\delta})\frac{{\Gamma}(d+{\delta})}{{\Gamma}(d+3+{\delta}+{\delta}/m)}.\end{array}$
(3.38)

Then, by (3.10) and (3.38), we get

$\begin{array}[]{lll}\displaystyle\sum_{d=m}^{r-1}p_{d}b_{d}^{(r)}=c({\delta})(-1)^{r}{\Gamma}(r+{\delta})\displaystyle\sum_{d=m}^{r-1}(-1)^{d}\frac{1}{(r-d)!{\Gamma}(d+3+{\delta}+{\delta}/m)}\\\
=c({\delta})\frac{(-1)^{r}{\Gamma}(r+{\delta})}{r+2+{\delta}+{\delta}/m}\\\
\displaystyle\sum_{d=m}^{r-1}(-1)^{d}\Bigg{[}\frac{1}{(r-d)!{\Gamma}(d+2+{\delta}+{\delta}/m)}+\frac{1}{(r-d-1)!{\Gamma}(d+3+{\delta}+{\delta}/m)}\Bigg{]}\\\
=c({\delta})\frac{(-1)^{r}{\Gamma}(r+{\delta})}{r+2+{\delta}+{\delta}/m}\Big{[}(-1)^{m}\frac{1}{(r-m)!{\Gamma}(m+2+{\delta}+{\delta}/m)}+(-1)^{r-1}\frac{1}{{\Gamma}(r+2+{\delta}+{\delta}/m)}\Big{]},\end{array}$
(3.39)

where at the last step we used the telescoping property of the sum. Thus by
(3.39), we get

$\begin{array}[]{lll}b_{m}^{(r)}-\dfrac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r}b_{d}^{(r)}p_{d}=b_{m}^{(r)}-\frac{r+{\delta}}{2m+{\delta}}p_{r}-\frac{r+{\delta}}{2m+{\delta}}\displaystyle\sum_{d=m}^{r-1}b_{d}^{(r)}p_{d}\\\
=(-1)^{r-m}\frac{{\Gamma}(r+{\delta})}{(r-m)!{\Gamma}(m+{\delta})}-(-1)^{r-m}\frac{(r+{\delta})(2+{\delta}/m){\Gamma}(r+{\delta})}{(2m+{\delta})(r-m)!{\Gamma}(m+{\delta})(r+2+{\delta}+{\delta}/m)}\\\
=\frac{(-1)^{r-m}{\Gamma}(r+{\delta})}{(r-m)!{\Gamma}(m+{\delta})}\frac{r+2+{\delta}-r/m}{r+2+{\delta}+{\delta}/m}.\end{array}$
(3.40)

###  3.1 First condition of Theorem 3.1

In this subsection we check that the first condition of Theorem 3.1 holds. We
know from (3.36) that

$sG_{s,s,j}(r,l)\rightarrow a^{(j)}(r,l),$ (3.41)

almost surely as $s\rightarrow\infty$. From definition (3.16) we have

$G_{s,h,j}(r,l)=\frac{a_{h,j}^{(r)}a_{h,j}^{(l)}}{sa_{s,j}^{(r)}a_{s,j}^{(l)}}hG_{h,h,j}(r,l).$
(3.42)

From the regular variation property (3.12), the function

$h\to u(h):=a_{h,j}^{(r)}a_{h,j}^{(l)}hG_{h,h,j}(r,l)$ (3.43)

is regularly varying with index $(r+l+2{\delta})/(2m+{\delta})$. Therefore,
from (3.42) and Karamata’s theorem on integration of regularly varying
functions [13, page 25]

$\begin{array}[]{lll}\displaystyle\sum_{j=0}^{m-1}V_{s,j}(r,l)=\displaystyle\sum_{j=0}^{m-1}\displaystyle\sum_{h=k+1}^{s}G_{s,h,j}(r,l)=\displaystyle\sum_{j=0}^{m-1}\displaystyle\sum_{h=k+1}^{s}\frac{a_{h,j}^{(r)}a_{h,j}^{(l)}}{sa_{s,j}^{(r)}a_{s,j}^{(l)}}hG_{h,h,j}(r,l)\\\
\displaystyle\sum_{j=0}^{m-1}\displaystyle\sum_{h=k+1}^{s}\frac{u(h)}{sa_{s,j}^{(r)}a_{s,j}^{(l)}}\sim\displaystyle\sum_{j=0}^{m-1}\frac{u(s)}{\Big{(}\frac{r+l+2{\delta}}{2m+{\delta}}+1\Big{)}a_{s,j}^{(r)}a_{s,j}^{(l)}}\sim
a(r,l)\frac{2m+{\delta}}{r+l+2m+3{\delta}},\end{array}$ (3.44)

for $r,l=m,m+1,...,k$.

This verifies the first condition the martingale central limit theorem of
Proposition 3.1 (with each $A_{n}$ being a $(k-m+1)\times(k-m+1)$ identity
matrix).

###  3.2 Second condition of Theorem 3.1

Next we show that the second condition of Theorem 3.1 holds as well. By (3.4),
we deduce that

$|(N_{l}(s,i)-{\mathbb{E}}[N_{l}(s,i)])-(N_{l}(s,i-1)-{\mathbb{E}}[N_{l}(s,i-1)])|\leq
2\quad\hbox{for all }l.$ (3.45)

Hence the events $\\{|X_{s,i,j,h}^{(l)}|>\varepsilon\\}$ are empty for all
$l$, for all $h\leq s$ and for all $j\leq i$ as $s\rightarrow\infty$, indeed

$\begin{array}[]{lll}\\{|X_{s,i,j,h}^{(l)}|>\varepsilon\\}&=&\\{|M_{h,j}^{(l)}-M_{h,j-1}^{(l)}|>a_{s,j}^{(l)}s^{1/2}\varepsilon\\}\\\
&=&\Big{\\{}|a_{h,j}^{(l)}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(N_{t}(h,j)-{\mathbb{E}}[N_{t}(h,j)])-a_{h,j-1}^{(l)}\\\
&&\times\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}(N_{t}(h,j-1)-{\mathbb{E}}[N_{t}(h,j-1)])>a_{s,j}^{(l)}s^{1/2}\varepsilon\Big{\\}}\\\
&\subset&\Big{\\{}C_{1}h^{\frac{l+{\delta}}{2m+{\delta}}}>\varepsilon
s^{\frac{l+{\delta}}{2m+{\delta}}}s^{1/2}\Big{\\}}\subset\Big{\\{}C_{1}>\varepsilon
s^{1/2}\Big{\\}},\end{array}$ (3.46)

where $C_{1}>0$ is a large constant independent of $s$. Thus we obtain

${\mathds{1}}_{\\{|X_{s,i,j,h}^{(l)}|>\varepsilon\\}}\leq{\mathds{1}}_{\\{C_{1}>\varepsilon
s^{1/2}\\}}\rightarrow 0,$ (3.47)

as $s\rightarrow\infty$. This implies that the second condition of Proposition
3.1 holds.

###  3.3 The limit process for degree counts

We conclude that

$\Bigg{(}\frac{1}{s^{1/2}}\displaystyle\sum_{j=m}^{k}b_{j}^{(k)}(N_{j}(s,i)-{\mathbb{E}}[N_{j}(s,i)]),\
k\geq m\Bigg{)}\Rightarrow(Y_{k},\ k\geq m)$ (3.48)

in $\mathbb{R}^{{\mathbb{N}}}$, where $(Y_{k},\ k=m,m+1,...)$ is a mean zero
Gaussian process with covariance function $R_{Y}$ given by

$R_{Y}(r,l)=\frac{2m+{\delta}}{r+l+2m+3{\delta}}a(r,l),\quad r,l\geq m.$
(3.49)

We use this covariance function to define the $(k-m+1)\times(k-m+1)$ matrix
with entries

$R_{Y,k}=(R_{Y}(r,l),\ m\leq r\leq k,m\leq l\leq k).$ (3.50)

The convergence in (3.48) means that for all $k=m,m+1,...$

$C_{k}\Big{(}\frac{N_{r}(s,i)-{\mathbb{E}}[N_{r}(s,i)]}{s^{1/2}},\
r=m,...,k\Big{)}^{T}\Rightarrow(Y_{r},\ r=m,...,k),$ (3.51)

where $C_{k}$ is a $(k-m+1)\times(k-m+1)$ matrix with entries

$c_{r,l}=\begin{cases}b_{l}^{(r)}&\hbox{ if }l\leq r,\\\ 0&\hbox{ if
}l>r.\end{cases}$ (3.52)

We are able to conclude that

$\Big{(}\frac{N_{r}(s,i)-{\mathbb{E}}[N_{r}(s,i)]}{s^{1/2}},\
r=m,...,k\Big{)}\Rightarrow D_{k}(Y_{r},\ r=m,..,k)^{T},$ (3.53)

and the covariance matrix of the limiting Gaussian vector is given by

$\Sigma_{k}=D_{k}R_{Y,k}D_{k}^{T}.$ (3.54)

Using the following identity,

$\displaystyle\sum_{t=l}^{r}x^{t-l}b_{t}^{(r)}b_{l}^{(t)}=b_{l}^{(r)}(1+x)^{r-l},\
1\leq l\leq r,\ x\in{\mathbb{R}},$ (3.55)

it can be shown that $D_{k}=C_{k}^{-1}$ has entries

$d_{r,l}=\begin{cases}(-1)^{r-l}b_{l}^{(r)}&\hbox{ if }l\leq r,\\\ 0&\hbox{ if
}l>r.\end{cases}$ (3.56)

In order to facilitate the computation of the entries of the matrix
$\Sigma_{k}$, by (3.37) we can write $R_{Y,k}$ in the form

$\begin{array}[]{lll}R_{Y,k}&=&\displaystyle\sum_{q=0}^{\infty}h_{q}\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}R_{q,x}^{1}dx\\\
&&+\displaystyle\sum_{q=m}^{\infty}h_{q}\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}R_{q,x}^{2}dx,\end{array}$
(3.57)

where

$h_{q}=\begin{cases}-1&\hbox{ if }q=0,1,...m-1,\\\
\frac{q+{\delta}}{2m+{\delta}}p_{q}&\hbox{ if }q=m,m+1,...,\end{cases}$ (3.58)

and the matrix $R_{m,x}$ is the $(k-m+1)\times(k-m+1)$ matrix with entries

$\begin{array}[]{ll}R_{q,x}^{1}=(C_{q,x}^{1})^{T}(C_{q,x}^{1}),\\\
R_{q,x}^{2}=(C_{q,x}^{2})^{T}(C_{q,x}^{2}).\end{array}$ (3.59)

The vector $C_{q,x}^{1}$ has the entries

$C_{q,x}^{1}(l)=\begin{cases}\Big{(}b_{m}^{(l)}-\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Big{)}e^{-lx},\
l\geq m&\hbox{ if }q=0,\\\
\Big{(}\frac{l+{\delta}}{2m+{\delta}}\displaystyle\sum_{t=m}^{l}b_{t}^{(l)}p_{t}\Big{)}e^{-lx},\
l\geq m&\hbox{ if }q=1,...,m-1,\\\
(b_{m}^{(l)}-b_{q}^{(l)}+b_{q+1}^{(l)})e^{-lx},\ l\geq m&\hbox{ if }q\geq
m.\end{cases}$ (3.60)

and

$C_{q,x}^{2}(l)=\begin{cases}0&\hbox{ if }q=0,1,...,m-1,\\\
\sqrt{m-1}(b_{q+1}^{(l)}-b_{q}^{(l)})e^{-lx}&\hbox{ if }q\geq m.\end{cases}$
(3.61)

Therefore

$\begin{array}[]{lll}\Sigma_{k}&=&\displaystyle\sum_{q=0}^{\infty}h_{q}\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}(D_{k}(C_{q,x}^{1})^{T})C_{q,x}^{1}D_{k}^{T}dx\\\
&&+\displaystyle\sum_{q=m}^{\infty}h_{q}\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}(D_{k}(C_{q,x}^{2})T)C_{q,x}^{2}D_{k}^{T}dx.\end{array}$
(3.62)

We compute separately the terms in the sum for $q<m$ and $q\geq m$. We begin
with $C_{q,x}^{1}$. For $q\geq m$ and $l=m,m+1,...$, since $b_{j}^{(k)}=0$ for
all $j>k$, by (3.55) and (3.56) we have

$\begin{array}[]{lll}(D_{k}(C_{q,x}^{1})^{T})(l)\\\
=\displaystyle\sum_{t=m}^{l}(-1)^{l-t}b_{t}^{(l)}(b_{m}^{(t)}-b_{q}^{(t)}+b_{q+1}^{(t)})e^{-tx}\\\
=\displaystyle\sum_{t=m}^{l}(-1)^{l-t}b_{t}^{(l)}b_{m}^{(t)}e^{-tx}-\displaystyle\sum_{t=q}^{l}(-1)^{l-t}b_{t}^{(l)}b_{q}^{(t)}e^{-tx}+\displaystyle\sum_{t=q+1}^{l}(-1)^{l-t}b_{t}^{(l)}b_{q+1}^{(t)}e^{-tx}\\\
=(-1)^{l-m}e^{-mx}b_{m}^{(l)}(1-e^{-x})^{l-m}-(-1)^{l-q}e^{-qx}b_{q}^{(l)}(1-e^{-x})^{l-q}\\\
\quad+(-1)^{l-q-1}e^{-(q+1)x}b_{q+1}^{(l)}(1-e^{-x})^{l-q-1}.\end{array}$
(3.63)

Therefore, for $q\geq m$ and $r,l=m,m+1,...$, we have

$\begin{array}[]{lll}(D_{k}(C_{q,x}^{1})^{T})(l)(D_{k}(C_{q,x}^{1})^{T})(r)\\\
=(-1)^{l+r}b_{m}^{(l)}b_{m}^{(r)}e^{-2mx}(1-e^{-x})^{l+r-2m}\\\
\quad-(-1)^{l+r-m-q}(b_{m}^{(l)}b_{q}^{(r)}+b_{m}^{(r)}b_{q}^{(l)})e^{-(m+q)x}(1-e^{-x})^{l+r-m-q}\\\
\quad+(-1)^{l+r-q-m-1}(b_{m}^{(l)}b_{q+1}^{(r)}+b_{m}^{(r)}b_{q+1}^{(l)})e^{-(m+q+1)x}(1-e^{-x})^{l+r-m-q-1}\\\
\quad+(-1)^{l+r}b_{q}^{(l)}b_{q}^{(r)}e^{-2qx}(1-e^{-x})^{r+l-2q}\\\
\quad+(-1)^{l+r}(b_{q}^{(l)}b_{q+1}^{(r)}+b_{q}^{(r)}b_{q+1}^{(l)})e^{-(2q+1)x}(1-e^{-x})^{l+r-2q-1}\\\
\quad+(-1)^{l+r}b_{q+1}^{(l)}b_{q+1}^{(r)}e^{-2(q+1)x}(1-e^{-x})^{l+r-2q-2}\\\
:=\displaystyle\sum_{t=1}^{6}\theta_{q,x}^{(t)}(r,l).\end{array}$ (3.64)

We have

$\begin{array}[]{lll}\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}\theta_{q,x}^{(1)}(r,l)dx\\\
=(-1)^{l+r}(2m+{\delta})b_{m}^{(l)}b_{m}^{(r)}\displaystyle\int_{0}^{+\infty}e^{-(4m+3{\delta})x}(1-e^{-x})^{l+r-2m}dx\\\
=(-1)^{l+r}(2m+{\delta})b_{m}^{(l)}b_{m}^{(r)}B(4m+3{\delta},l+r-2m+1)\\\
=(-1)^{l+r}(2m+{\delta})b_{m}^{(l)}b_{m}^{(r)}\frac{{\Gamma}(4m+3{\delta})(l+r-2m)!}{{\Gamma}(l+r+2m+1+3{\delta})},\end{array}$
(3.65)

where

$B(\alpha,\beta)=\frac{{\Gamma}(\alpha){\Gamma}(\beta)}{{\Gamma}(\alpha+\beta)}.$
(3.66)

is the Beta function. Similarly,

$\displaystyle\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}\theta_{q,x}^{(2)}(r,l)dx$
$\displaystyle=(-1)^{l+r-m-q+1}(2m+{\delta})(b_{m}^{(l)}b_{q}^{(r)}+b_{m}^{(r)}b_{q}^{(l)})$
$\displaystyle\times\frac{{\Gamma}(3m+q+3{\delta})(l+r-m-q)!}{{\Gamma}(l+r+2m+1+3{\delta})},$
(3.67)
$\displaystyle\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}\theta_{q,x}^{(3)}(r,l)dx$
$\displaystyle=(-1)^{l+r-m-q-1}(2m+{\delta})(b_{m}^{(l)}b_{q+1}^{(r)}+b_{m}^{(r)}b_{q+1}^{(l)})$
$\displaystyle\frac{{\Gamma}(3m+q+1+3{\delta})(l+r-m-q-1)!}{{\Gamma}(l+r+2m+1+3{\delta})},$
(3.68)
$\displaystyle\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}\theta_{q,x}^{(4)}(r,l)dx$
$\displaystyle=(-1)^{r+l}(2m+{\delta})b_{q}^{(l)}b_{q}^{(r)}\frac{{\Gamma}(2q+2m+3{\delta})(r+l-2q)!}{{\Gamma}(r+l+2m+1+3{\delta})},$
(3.69)
$\displaystyle\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}\theta_{q,x}^{(5)}(r,l)dx$
$\displaystyle=(-1)^{l+r}(2m+{\delta})(b_{q}^{(l)}b_{q+1}^{(r)}+b_{q}^{8r)}b_{q+1}^{(l)})\frac{{\Gamma}(2q+2m+1+3{\delta})(l+r-2q-1)!}{{\Gamma}(l+r+2m+1+3{\delta})},$
(3.70)
$\displaystyle\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}\theta_{q,x}^{(6)}(r,l)dx$
$\displaystyle=(-1)^{l+r}(2m+{\delta})b_{q+1}^{(l)}b_{q+1}^{(r)}\frac{{\Gamma}(2q+2m+2+3{\delta})(l+r-2q-2)!}{{\Gamma}(l+r+2m+1+3{\delta})}.$
(3.71)

By (3.40) and (3.56), for $q=0$ and $l=m,m+1,...$ we have

$\begin{array}[]{lll}(D_{k}(C_{0,x}^{1})^{T})(l)\\\
=\displaystyle\sum_{t=m}^{l}(-1)^{l-t}b_{t}^{(l)}\Bigg{(}b_{m}^{(t)}-\frac{t+{\delta}}{2m+{\delta}}\displaystyle\sum_{h=m}^{t}b_{h}^{(t)}p_{h}\Bigg{)}e^{-tx}\\\
=\frac{{\Gamma}(l+{\delta})}{{\Gamma}(m+{\delta})}\displaystyle\sum_{t=m}^{l}(-1)^{t-m}e^{-tx}\frac{t+2+{\delta}-t/m}{(t-m)!(l-t)!(t+2+{\delta}+{\delta}/m)}.\end{array}$
(3.72)

Therefore, by (3.72), for $r,l\geq m$ and $q=0$, we obtain

$\begin{array}[]{lll}\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}(D_{k}(C_{0,x}^{1})^{T})(r)(C_{0,x}^{1}D_{k}^{T})(l)dx\\\
=(2m+{\delta})\frac{{\Gamma}(l+{\delta}){\Gamma}(r+{\delta})}{({\Gamma}(m+{\delta}))^{2}}\displaystyle\sum_{t_{1}=m}^{l}\displaystyle\sum_{t_{2}=m}^{r}(-1)^{t_{1}+t_{2}}\\\
\quad\times\frac{(t_{1}+t_{2}+2m+3{\delta})^{-1}(t_{1}+2+{\delta}-t_{1}/m)(t_{2}+2+{\delta}-t_{2}/m)}{(t_{1}-m)!(t_{2}-m)!(l-t_{1})!(r-t_{2})!(t_{1}+2+{\delta}+{\delta}/m)(t_{2}+2+{\delta}+{\delta}/m)}\end{array}$
(3.73)

Consider now the case $q=1,...,m-1$ and $l=m,m+1,...$. By (3.40) and (3.56),
we have

$\begin{array}[]{lll}(D_{k}(C_{q,x}^{1})^{T})(l)\\\
&=&\displaystyle\sum_{t=m}^{l}(-1)^{l-t}b_{t}^{(l)}\Bigg{(}\frac{t+{\delta}}{2m+{\delta}}\displaystyle\sum_{h=m}^{t}b_{h}^{(t)}p_{h}\Biggl{)}e^{-tx}\\\
&=&{\Gamma}(l+{\delta})\displaystyle\sum_{t=m}^{l}\frac{e^{-tx}}{(l-t)!{\Gamma}(t+{\delta})}\\\
&&\times\Bigg{(}b_{m}^{(t)}-(-1)^{t-m}\frac{{\Gamma}(t+{\delta})(t+2+{\delta}-t/m)}{(t-m)!{\Gamma}(m+{\delta})(t+2+{\delta}+{\delta}/m)}\Bigg{)}.\end{array}$
(3.74)

Therefore, by (3.74), for $r,l\geq m$ and $q=1,...,m-1$, we have

$\begin{array}[]{lll}\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}(D_{k}(C_{q,x}^{1})^{T})(r)(C_{q,x}^{1}D_{k}^{T})(l)dx\\\
=(2m+{\delta}){\Gamma}(l+{\delta}){\Gamma}(r+{\delta})\displaystyle\sum_{t_{1}=m}^{l}\displaystyle\sum_{t_{2}=m}^{r}\frac{(t_{1}+t_{2}+2m+3{\delta})^{-1}}{(l-t_{1})!{\Gamma}(t_{1}+{\delta})(r-t_{2})!{\Gamma}(t_{2}+{\delta})}\\\
\quad\times\Bigg{(}b_{m}^{(t_{1})}-(-1)^{t_{1}-m}\frac{{\Gamma}(t_{1}+{\delta})(t_{1}+2+{\delta}-t_{1}/m)}{(t_{1}-m)!{\Gamma}(m+{\delta})(t_{1}+2+{\delta}+{\delta}/m)}\Bigg{)}\\\
\quad\times\Bigg{(}b_{m}^{(t_{2})}-(-1)^{t_{2}-m}\frac{{\Gamma}(t_{2}+{\delta})(t_{2}+2+{\delta}-t_{2}/m)}{(t_{2}-m)!{\Gamma}(m+{\delta})(t_{2}+2+{\delta}+{\delta}/m)}\Bigg{)}.\\\
\end{array}$ (3.75)

By using (3.10), we deduce

$\begin{array}[]{lll}\displaystyle\int_{0}^{+\infty}(2m+{\delta})e^{-(2m+3{\delta})x}(D_{k}(C_{q,x}^{1})^{T})(r)(C_{q,x}^{1}D_{k}^{T})(l)dx\\\
\quad=\frac{(2m+{\delta}){\Gamma}(l+{\delta}){\Gamma}(r+{\delta})}{m^{2}({\Gamma}(m+{\delta}))^{2}}\displaystyle\sum_{t_{1}=m}^{l}\displaystyle\sum_{t_{2}=m}^{r}(-1)^{t_{1}+t_{2}}\\\
\quad\quad\times\frac{({\delta}+t_{1})({\delta}+t_{2})(t_{1}+t_{2}+2m+3{\delta})^{-1}}{(t_{1}-m)!(t_{2}-m)!(l-t_{1})!(r-t_{2})!(t_{1}+2+{\delta}+{\delta}/m)(t_{2}+2+{\delta}+{\delta}/m)}\end{array}$
(3.76)

Now we investigate the behavior of the term $C_{q,x}^{2}$. For $q=m,m+1,...,$,
using (3.55) we get

$\begin{array}[]{lll}(D_{k}(C_{q,x}^{2})^{T})(l)&=&\displaystyle\sum_{t=m}^{l}\sqrt{m-1}(-1)^{l-t}b_{t}^{(l)}(b_{q+1}^{(t)}-b_{q}^{(t)})e^{-tx}\\\
&=&\sqrt{m-1}\Big{(}(-1)^{q+1-l}e^{-(q+1)x}b_{q+1}^{(l)}(1-e^{-x})^{l-q-1}\\\
&&-(-1)^{q-l}e^{-qx}b_{q}^{(l)}(1-e^{-x})^{l-q}.\end{array}$ (3.77)

Thus we obtain

$\begin{array}[]{lll}(D_{k}(C_{q,x}^{2})^{T})(r)(D_{k}(C_{q,x}^{2})^{T})(l)\\\
=(m-1)\Big{(}(-1)^{r+l}e^{-2(q+1)x}b_{q+1}^{(l)}b_{q+1}^{(r)}(1-e^{-x})^{l+r-2q-2}\\\
\quad+(-1)^{r+l}e^{-2qx}b_{q}^{(l)}b_{q}^{(r)}(1-e^{-x})^{l+r-2q}\\\
\quad+(-1)^{r+l}e^{-(2q+1)x}(b_{q+1}^{(l)}b_{q}^{(r)}+b_{q}^{(l)}b_{q+1}^{(r)})(1-e^{-x})^{l+r-2q-1}\Big{)}\\\
=(m-1)(\theta_{q,x}^{(4)}+\theta_{q,x}^{(5)}+\theta_{q,x}^{(6)}),\end{array}$
(3.78)

where $\theta_{q,x}^{(i)}$, $i=4,5,6$, are defined in (3.64).

Using (3.65)-(3.73), (3.76) and (3.78), we conclude that the covariance
function $R_{Z}(r,l)$ of the limiting Gaussian process $(Z_{k},\ k=m,m+1,...)$
in (2.6) is given by (2.7).

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