# On asymptotic normality of the total progeny in the positive recurrent
Q-processes

Azam A. IMOMOV  Azam Abdurakhimovich Imomov
iii Karshi State University,
iii 17, Kuchabag st., 180100 Karshi city, Uzbekistan.  imomov_<EMAIL_ADDRESS>and  Zuhriddin A. NAZAROV  Zuhriddin A. Nazarov
iii V.I.Romanovskiy Institute of Mathematics,
iii Uzbekistan Academy of Sciences, Tashkent 100174, Uzbekistan.
<EMAIL_ADDRESS>Dedicated to the Fond Memory of our Great Ancestry

###### Abstract.

We examine the population growth system called Q-processes. This is defined by
the Galton-Watson Branching system conditioned on non-extinction of its
trajectory in the remote future. In this paper we observe the total progeny up
to time $n$ in the Q-process. By analogy with branching systems, this variable
is of great interest in studying the deep properties of the Q-process. We find
that the sum total progeny as a random variable approximates the standard
normal distribution function under a second moment assumption for the initial
Galton-Watson system offspring law. We estimate the speed rate of this
approximation.

###### Key words and phrases:

Branching system, Q-process, Markov chain, generating function, transition
probabilities, invariant distribution, extinction time, total progeny,
positive recurrent, central limit theorem, law of large numbers.

###### 2010 Mathematics Subject Classification:

Primary 60J80; Secondary 60J85

© 2023 Imomov A.A, Nazarov Z.A

##  1\. Introduction and main results

In the general theory of random processes models of stochastic branching
systems are particularly important. Nowadays, there is great interest in these
models. The creation of the theory of branching models is related to the
possibility of estimating the survival probability of the population of
monotypic individuals. The discrete-time simple branching process model was
introduced by Francis Galton in 1889 as a mathematical model for the
population family growth is now called the Galton-Watson Branching (GWB)
system; see [1], [2], [4], [7], [8], [9] and [12]. GWB models play a
fundamental role in both the theory and applications of stochastic processes.
Among the random trajectories of branching systems, there are those that
continue a long time. In the case of the GWB model, the class of such
trajectories forms another stochastic model called Q-process; see [2] and [6].
In the case of continuous-time Markov branching systems, an analogous model
called the _Markov Q-process_ , was first introduced in [5].

Let $\left\\{{Z(n),n\in{\mathbb{N}}_{0}}\right\\}$ GWB system with branching
rates $\left\\{{p_{k},k\in\mathbb{N}_{0}}\right\\}$, where
$\mathbb{N}_{0}=\\{0\\}\cup\mathbb{N}$ and $\mathbb{N}=\\{1,2,\ldots\\}$, the
variable $Z(n)$ denote the population size at the moment $n$ in the system.
The evolution of the system occurs according to the following mechanism. Each
individual lives a unit length life time and then gives $k\in\mathbb{N}_{0}$
descendants with probability $p_{k}$. This process is a reducible,
homogeneous-discrete-time Markov chain with a state space consisting of two
classes: $\mathcal{S}_{0}=\\{0\\}\cup\mathcal{S}$, where $\\{0\\}$ is
absorbing state, and $\mathcal{S}\subset\mathbb{N}$ is the class of possible
essential communicating states. Throughout the paper assume that $p_{0}>0$ and
$p_{0}+p_{1}>0$ which called the Schröder case. We suppose that
$p_{0}+p_{1}<1$ and $m:=\sum_{k\in\mathcal{S}}{kp_{k}}<\infty$.

Considering transition probabilities

$P_{ij}(n):=\textsf{P}\left\\{{Z(n+k)=j}\bigm{|}Z(k)=i\right\\}\qquad{for\;any}\quad
k\in\mathbb{N}_{0}$

we observe that the corresponding probability generating function (GF)

$\sum_{k\in\mathcal{S}_{0}}{P_{ij}(n)s^{k}}=\bigl{[}{f_{n}(s)}\bigr{]}^{i},$
(1.1)

where $f_{n}(s):=\sum_{k\in\mathcal{S}_{0}}{\textsf{p}_{k}(n)s^{k}}$, therein
$\textsf{p}_{k}(n):=P_{1k}(n)$ and, in the same time $f_{n}(s)$ is $n$-fold
iteration of the offspring GF

$f(s):=\sum_{k\in\mathcal{S}_{0}}{p_{k}s^{k}}.$

Needless to say that $f_{n}(0)=\textsf{p}_{0}(n)$ is a vanishing probability
of the system initiated by one individual. Note that this probability tends as
$n\to\infty$ monotonously to $q$, which called an extinction probability of
the system, i.e. $\lim\nolimits_{n\to\infty}\textsf{p}_{0}(n)=q$; see [2].

The extinction probability

  * •

$q=1$ if $m\leq 1$;

  * •

$q<1$ if $m>1$.

Based on this, according to the values of the parameter $m$, the system is
called

  * •

sub-critical if $m<1$;

  * •

critical if $m=1$;

  * •

super-critical if $m>1$.

Further we are dealing with the GWB system conditioned on the event
$\left\\{n<\mathcal{H}<\infty\right\\}$, where $\mathcal{H}$ is an extinction
time of the system, i.e.
$\mathcal{H}:=\min\left\\{{n\in\mathbb{N}:Z(n)=0}\right\\}$. Let
$\textsf{P}_{i}\bigl{\\{}{*}\bigr{\\}}:=\textsf{P}\left\\{{{\ast}\bigm{|}{Z(0)=i}}\right\\}$
and define conditioned probability measure

$\textsf{P}_{i}^{\mathcal{H}(n+k)}\\{*\\}:=\textsf{P}_{i}\left\\{*\bigm{|}n+k<{\mathcal{H}<\infty}\right\\}\qquad{for\;any}\quad
k\in\mathbb{N}.$

In [2, p. 58] proved, that

$\mathcal{Q}_{ij}(n):=\lim_{k\to\infty}\textsf{P}_{i}^{{\mathcal{H}}(n+k)}\bigl{\\{}Z(n)=j\bigr{\\}}={{jq^{j-i}}\over{i\beta^{n}}}P_{ij}(n),$
(1.2)

where $\beta:=f^{\prime}(q)$. Observe that
${\sum_{j\in{\mathbb{N}}}{\mathcal{Q}_{ij}(n)}}=1$ for each
$i\in{\mathbb{N}}$. Thus, the probability measure $\mathcal{Q}_{ij}(n)$ can
determine a new population growth system with the state space
$\mathcal{E}\subset\mathbb{N}$ which we denote by
$\left\\{W(n),n\in{\mathbb{N}}_{0}\right\\}$. This is a discrete-homogeneous-
time irreducible Markov chain defined in the book [2, p. 58] and called the
Q-process. Undoubtedly $W(0)\mathop{=}\limits^{d}Z(0)$ and transition
probabilities

${\mathcal{Q}}_{ij}(n):=\textsf{P}\left\\{W(n)=j\Bigm{|}{W(0)=i}\right\\}=\textsf{P}_{i}\left\\{Z(n)=j\Bigm{|}{\mathcal{H}}=\infty\right\\},$

so that the Q-process can be interpreted as a “long-living” GWB system.

Put into consideration a GF

$w_{n}^{(i)}(s):=\sum\limits_{j\in{\mathcal{E}}}{{\mathcal{Q}}_{ij}(n)s^{j}}.$

Then from (1.1) and (1.2) we obtain

$w_{n}^{(i)}(s)=\left[{{\frac{f_{n}(qs)}{q}}}\right]^{i-1}\cdot w_{n}(s),$
(1.3)

where the GF
$w_{n}(s):=w_{n}^{(1)}(s)=\textsf{E}\left[{s^{W(n)}\bigm{|}{W(0)=1}}\right]$
has a form of

$w_{n}(s)=s{\frac{f^{\prime}_{n}(qs)}{\beta^{n}}}\quad{for\;all}\quad
n\in{\mathbb{N}}.$ (1.4)

Using iterations for $f(s)$ in (1.3) leads to the following functional
equation:

$w_{n+1}^{(i)}(s)={\frac{w(s)}{f_{q}(s)}}w_{n}^{(i)}\bigl{(}{f_{q}(s)}\bigr{)},$
(1.5)

where $w(s):=w_{1}(s)$ and $f_{q}(s)={f(qs)\big{/}q}$. Thus, Q-process is
completely defined by setting the GF

$w(s)=s{\frac{f^{\prime}(qs)}{\beta}}\raise 1.5pt\hbox{.}$ (1.6)

An evolution of the Q-process is in essentially regulated by the structural
parameter $\beta>0$. In fact, as it has been shown in [2, p. 59, Theorem 2],
that

  * •

$\mathcal{E}$ is positive recurrent if $\beta<1$;

  * •

$\mathcal{E}$ is transient if $\beta=1$.

On the other hand, it is easy to be convinced that positive recurrent case
$\beta<1$ of Q-process is in a definition character of the non-critical case
$m\neq{1}$ of the initial GWB system. Note that $\beta\leq{1}$ and nothing
but.

In this paper we deal with the positive recurrent case assuming that first
moment $\alpha:=w^{\prime}(1-)$ be finite. Then differentiating (1.6) on the
point $s=1$ we obtain $\alpha=1+\gamma_{q}\cdot\left({1-\beta}\right)$, where

$\gamma_{q}:={\frac{qf^{\prime\prime}(q)}{\beta\left(1-\beta\right)}}\raise
1.5pt\hbox{.}$

It follows from (1.3) and (1.4) that

${\textsf{E}}_{i}W(n):=\textsf{E}\left[{W(n)\Bigm{|}{W(0)=i}}\right]=\left({i-1}\right)\beta^{n}+\textsf{E}W(n),$

where $\textsf{E}W(n)=1+\gamma_{q}\cdot\bigl{(}{1-\beta^{n}}\bigr{)}$.

It is obvious, that when initial GWB system is sub-critical, then the
condition $\alpha<\infty$ is this is equivalent to that
$f^{\prime\prime}(1-)<\infty$. Further we everywhere will be accompanied by
this condition by default.

Our purpose is to investigate asymptotic properties of a random variable

$S_{n}=W(0)+W(1)+\cdots+W(n-1),$

denoting the total number of individuals that have existed up to the $n$-th
generation in Q-process. By analogy with branching systems, this variable is
of great interest in studying the deep properties of the Q-process. For
details on the total progeny in GWB systems and related models results, see
e.g. [8], [9], [10], [11].

Throughout this paper we will use famous Landau symbols ${o}$, ${\mathcal{O}}$
and ${\mathcal{O}^{\ast}}$ to describe kinds of bounds on asymptotic varying
rates of positive functions $f(x)$ and $g(x)$. for for all large enough values
of at infinity. So, $f={o}(g)$ means that $\lim_{x}{f(x)}\big{/}{g(x)}=0$, and
we write $f={\mathcal{O}}(g)$ if $\limsup_{x}{f(x)}\big{/}{g(x)}<\infty$ and
also we write $f={\mathcal{O}^{\ast}}(g)$ if the ratio ${f(x)}\big{/}{g(x)}$
has a positive explicit limit. i.e. $\lim_{x}{f(x)}\big{/}{g(x)}=C<\infty$.
Moreover, $f\sim(g)$ means that $\lim_{x}{f(x)}\big{/}{g(x)}=1$.

Our main results are analogues of Central Limit Theorem and Law of Large
Numbers for $S_{n}$. Let $\mathcal{N}\left({0,\sigma^{2}}\right)$ be a normal
distributed random variable with the zero mean and the finite variance
$\sigma^{2}$ and ${\Phi}_{0,\sigma^{2}}(x)$ is its distribution function.

######  Theorem 1.

Let $\beta<1$ and $\alpha<\infty$. Then there exists a positive real-valued
sequence $\mathcal{K}_{n}$ such that
$\mathcal{K}_{n}=\mathcal{O}^{\ast}\left(\sqrt{n}\right)$ and

$\frac{S_{n}-\textsf{E}S_{n}}{\mathcal{K}_{n}}\buildrel\textsf{P}\over{\longrightarrow}\mathcal{N}\left({0,\sigma^{2}}\right)\qquad{as}\quad
n\to\infty,$

where the symbol “$\buildrel\textsf{P}\over{\longrightarrow}$” means the
convergence in probability.

######  Theorem 2.

Let $\beta<1$ and $\alpha<\infty$. Then there exists slowly varying function
at infinity $\mathcal{L}(\ast)$ such that

$\left|\textsf{P}\left\\{\frac{S_{n}-\textsf{E}S_{n}}{\mathcal{K}_{n}}<x\right\\}-{\Phi}_{0,\sigma^{2}}(x)\right|\leq{\frac{\mathcal{L}(n)}{n^{1/4}}}$

uniformly in $x$.

Let ${I_{a}}$ be a degenerate distribution concentrated at the point $a$, i.e.

${I_{a}(B)}=\left\\{\begin{array}[]{l}1\qquad\hfill\text{if}\quad x\in{B},\\\
\\\ 0\qquad\hfill\text{if}\quad x\notin{B}.\end{array}\right.$

######  Theorem 3.

Let $\beta<1$ and $\alpha<\infty$. Then

$\frac{S_{n}}{n}\buildrel\textsf{P}\over{\longrightarrow}{1+\gamma_{q}}\qquad{as}\quad
n\to\infty.$

Moreover there exists slowly varying function at infinity
${\mathcal{L}_{\gamma}(\ast)}$ such that

$\left|\textsf{P}\left\\{\frac{S_{n}}{n}<x\right\\}-I_{1+\gamma_{q}}(x)\right|\leq{\frac{\mathcal{L}_{\gamma}(n)}{\sqrt{n}}}$

uniformly in $x$, where

${I_{1+\gamma_{q}}(x)}=\left\\{\begin{array}[]{l}0\qquad\hfill\text{if}\quad
x\leq 1+\gamma_{q},\\\ \\\ 1\qquad\hfill\text{if}\quad
x>1+\gamma_{q}.\end{array}\right.$

The rest of this paper is organized as follows. Section 2 provides auxiliary
statements that will be essentially used in the proof of our theorems. Section
3 is devoted to the proof of main results.

##  2\. Preliminaries

Further we need the joint GF of the variables $W(n)$ and $S_{n}$

$J_{n}(s;x)=\sum_{j\in{\mathcal{E}}}\sum_{l\in\mathbb{N}}\textsf{P}\left\\{W(n)=j,S_{n}=l\right\\}s^{j}x^{l}$

on a two-dimensional domain

$\mathbb{K}=\left\\{(s;x)\in\mathbb{R}^{2}:\;s\in[0,1],\;x\in[0,1],\;\sqrt{(s-1)^{2}+(x-1)^{2}}>0\right\\}.$

Due to the Markov nature of the Q-process, we see that the two-dimensional
one-step joint-transition probabilities

$\textsf{P}\left\\{W(n+1)=j,S_{n+1}=l\Bigm{|}W(n)=i,S_{n}=k\right\\}=\textsf{P}_{i}\left\\{W(1)=j,S_{1}=l\right\\}\delta_{l,i+k},$

where $\delta_{ij}$ is the Kronecker’s delta function:

${\delta_{ij}}=\left\\{\begin{array}[]{l}1\qquad\hfill\text{if}\quad i=j,\\\
\\\ 0\qquad\hfill\text{if}\quad i\neq{j}.\end{array}\right.$

Therefore, we have

$\displaystyle\textsf{E}_{i}\left[s^{W(n+1)}x^{S_{n+1}}\Bigm{|}S_{n}=k\right]$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{E}}\sum_{l\in\mathbb{N}}\textsf{P}_{i}\left\\{W(1)=j,S_{1}=l\right\\}\delta_{l,i+k}s^{j}x^{l}$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{E}}\textsf{P}_{i}\left\\{W(1)=j\right\\}s^{j}x^{i+k}=w^{(i)}(s)x^{i+k}.$

Next, using the formula of total probabilities, we obtain

$\displaystyle J_{n+1}(s;x)$ $\displaystyle=$
$\displaystyle\textsf{E}\left[\textsf{E}\left[s^{W(n+1)}x^{S_{n+1}}\Bigm{|}W(n),S_{n}\right]\right]=\textsf{E}\left[w^{W(n)}(s)x^{W(n)+S_{n}}\right]$
$\displaystyle=$
$\displaystyle\textsf{E}\left[\bigl{(}w(s)f_{q}(s)\bigr{)}^{W(n)-1}x^{W(n)+S_{n}}\right]=\frac{w(s)}{f_{q}(s)}\textsf{E}\left[\bigl{(}xf_{q}(s)\bigr{)}^{W(n)}x^{S_{n}}\right].$

In the last line we used formula (1.3). Thus we have

$J_{n+1}(s;x)=\frac{w(s)}{f_{q}(s)}J_{n}\bigl{(}xf_{q}(s);x\bigr{)}$ (2.1)

for $(s;x)\in\mathbb{K}$ and any $n\in\mathbb{N}$.

Using relation (2.1), we can now obtain an explicit expression for the GF
$J_{n}(s;x)$. Indeed, applying it consistently, taking into account (1.6) and,
after standard transformations, we have

$J_{n}(s;x)=\frac{s}{\beta^{n}}\frac{\partial H_{n}(s;x)}{\partial{s}}\raise
1.0pt\hbox{,}$ (2.2)

where the function $H_{n}(s;x)$ is defined for any $(s;x)\in\mathbb{K}$ by the
following recursive relations:

$\left\\{\begin{array}[]{l}H_{0}(s;x)=s;\\\ \\\
H_{n+1}(s;x)=xf_{q}\bigl{(}H_{n}(s;x)\bigr{)}.\end{array}\right.$ (2.3)

Since $\partial J_{n}(s;x)\bigl{/}\partial
x\Bigl{|}_{(s;x)=(1;1)}=\textsf{E}S_{n}$, from (2.2) and (2.3), we find that

$\textsf{E}S_{n}=(1+\gamma_{q})n-\gamma_{q}\frac{1-\beta^{n}}{1-\beta}\raise
1.0pt\hbox{.}$ (2.4)

######  Remark 1.

Needles to say that the GF $f_{q}(s)=f(qs)\bigl{/}q$ generates a sub-critical
GWB system. Denoting the population in this system as $Z_{q}(n)$, we define
the sum $V_{n}=\sum_{k=0}^{n-1}Z_{q}(k)$ which is a total progeny of
individuals that participated in the evolution of the system
$\left\\{Z_{q}(n),n\in\mathbb{N}_{0}\right\\}$, up to the $n$-th generation.
It is known that the GF of the joint distribution
$\bigl{(}Z_{q}(n),V_{n}\bigr{)}$ satisfies the recursive equation (2.3); see
[10, p. 126]. Thus, the function $H_{n}(s;x)$ is a two-dimensional GF for all
$n\in\mathbb{N}$ and $(s;x)\in\mathbb{K}$ and obeys to all properties of the
GF $\textsf{E}\left[s^{Z_{q}(n)}x^{V_{n}}\right]$.

By virtue of what said in Remark 1, in studying $H_{k}(s;x)$ we use the
properties of the GF $\textsf{E}\left[s^{Z_{q}(n)}x^{V_{n}}\right]$. Since the
system $\left\\{Z_{q}(n)\right\\}$ is sub-critical, it goes extinct with
probability $1$. Therefore, there exists a proper random variable
$V={\lim}_{n\to\infty}V_{n}$, which means the total number of individuals
participated in the whole evolution of the system. So

$h(x):=\textsf{E}x^{V}=\lim_{n\to\infty}\textsf{E}x^{V_{n}}=\lim_{n\to\infty}H_{n}(1;x)$

and, according to (2.3) it satisfies the functional equation

$h(x)=xf_{q}\bigl{(}h(x)\bigr{)}.$ (2.5)

Further, we note that

$\textsf{P}\bigl{\\{}Z_{q}(n)=0,V_{n}=k\bigr{\\}}=\textsf{P}\bigl{\\{}Z_{q}(n)=0,V=k\bigr{\\}}.$

Then, due to the monotonicity of the probabilistic GF, we find

$\textsf{P}\bigl{\\{}V=k\bigr{\\}}-\sum_{i\in\mathbb{N}}\textsf{P}\bigl{\\{}Z_{q}(n)=i,V_{n}=k\bigr{\\}}s^{i}\leq\textsf{P}\bigl{\\{}V=k,Z_{q}(n)>0\bigr{\\}}.$

Therefore, denoting

$R_{n}(s;x):=h(x)-H_{n}(s;x)$

for $(s;x)\in\mathbb{K}$, we have

$R_{n}(s;x)\leq\sum_{k\in\mathbb{N}}\textsf{P}\left\\{V=k,Z_{q}(n)>0\right\\}x^{k}=R_{n}(0;x).$

It is easy to see $R_{n}(0;x)\leq
R_{n}(0;1)=\textsf{P}\left\\{Z_{q}(n)>0\right\\}$. Then

$\bigl{|}R_{n}(s;x)\bigr{|}\leq\textsf{P}\left\\{Z_{q}(n)>0\right\\}\longrightarrow
0\qquad{as}\quad n\to\infty.$ (2.6)

On the other hand, due to the fact that $\left|h(x)\right|\leq 1$ and
$\left|H_{n}(s;x)\right|\leq 1$ we have

$\displaystyle R_{n}(s;x)$ $\displaystyle=$ $\displaystyle
x\left[f_{q}\bigl{(}h(x)\bigr{)}-f_{q}\bigl{(}H_{n-1}(s;x)\bigr{)}\right]$
$\displaystyle=$ $\displaystyle
x\textsf{E}\bigl{[}h(x)-H_{n-1}(s;x)\bigr{]}^{Z_{q}(n)}\leq\beta R_{n-1}(s;x)$

for all $(s;x)\in\mathbb{K}$. This implies that

$\bigl{|}R_{n}(s;x)\bigr{|}\leq\beta^{n-k}\bigl{|}R_{k}(s;x)\bigr{|}$ (2.7)

for any $n\in\mathbb{N}$ and $k=0,1,\dots,n$.

In what follows, where the function $R_{n}(s;x)$ will be used, we deal with
the domain $\mathbb{K}$, where this function does not vanish. By virtue of
(2.6), taking into account (2.3), (2.5), we obtain the asymptotic formula

$R_{n+1}(s;x)=xf_{q}^{\prime}\bigl{(}h(x)\bigr{)}R_{n}(s;x)-x\frac{f_{q}^{\prime\prime}\bigl{(}h(x)\bigr{)}+\eta_{n}(s;x)}{2}R_{n}^{2}(s;x),$
(2.8)

where $\left|\eta_{n}(s;x)\right|\to 0$ as $n\to\infty$ uniformly in
$(s;x)\in\mathbb{K}$. Since $R_{n}(s;x)\to 0$, it follows from (2.8) that

$R_{n}(s;x)=\frac{R_{n+1}(s;x)}{xf_{q}^{\prime}\bigl{(}h(x)\bigr{)}}(1+o(1))\qquad{as}\quad
n\to\infty.$

Using last equality, we transform (2.8) to the form

$R_{n+1}(s;x)=xf_{q}^{\prime}\bigl{(}h(x)\bigr{)}R_{n}(s;x)-\left[\frac{f_{q}^{\prime\prime}\bigl{(}h(x)\bigr{)}}{2f_{q}^{\prime}\bigl{(}h(x)\bigr{)}}+\varepsilon_{n}(s;x)\right]R_{n}(s;x)R_{n+1}(s;x)$

and, therefore

$\frac{u(x)}{R_{n+1}(s;x)}=\frac{1}{R_{n}(s;x)}+\upsilon(x)+\varepsilon_{n}(s;x),$
(2.9)

where

$u(x)=xf_{q}^{\prime}\bigl{(}h(x)\bigr{)}\qquad{and}\qquad\upsilon(x)=x\frac{f_{q}^{\prime\prime}\bigl{(}h(x)\bigr{)}}{2u(x)}$

and
$\sup_{(s;x)\in\mathbb{K}}\bigl{|}\varepsilon_{n}(s;x)\bigr{|}\leq\varepsilon_{n}\to
0$ as $n\to\infty$. By successively applying (2.9), we find the following
representation for $R_{n}(s;x)$:

$\frac{u^{n}(x)}{R_{n}(s;x)}=\frac{1}{R_{0}(s;x)}+\frac{\upsilon(x)\bigl{[}1-u^{n}(x)\bigr{]}}{1-u(x)}+\sum_{k=1}^{n}\varepsilon_{k}(s;x)u^{k}(x).$
(2.10)

In what follows, our discussions will essentially be based on formula (2.10).
Note that in the monograph [10, p. 136] this formula was stated for the
critical GWB system.

Now, for convenience, we write

$J_{n}(s;x)=s\prod_{k=0}^{n-1}\frac{xf_{q}^{\prime}\bigl{(}H_{k}(s;x)\bigr{)}}{\beta}$

which is a direct consequence of formulas (2.2) and (2.3). In our notation, it
is almost obvious that $T_{n}(x):=\textsf{E}x^{S_{n}}=J_{n}(1;x)$. Then it
follows that

$T_{n}(x)=\prod_{k=0}^{n-1}u_{k}(x),$ (2.11)

where

$u_{n}(x)=\frac{xf_{q}^{\prime}\bigl{(}h_{n}(x)\bigr{)}}{\beta}\raise
1.0pt\hbox{,}$

at that $h_{n}(x)=\textsf{E}x^{V_{n}}$ which satisfies a recurrence equation
$h_{n+1}(x)=xf_{q}\bigl{(}h_{n}(x)\bigr{)}$. Accordingly, the function
$\Delta_{n}(x):=h(x)-h_{n}(x)$ satisfies the inequality

$\bigl{|}\Delta_{n}(x)\bigr{|}\leq\beta^{n-k}\bigl{|}\Delta_{k}(x)\bigr{|}$
(2.12)

which is a consequence of (2.7). Successive application of the inequality
(2.12) gives

$\bigl{|}\Delta_{n}(x)\bigr{|}=\mathcal{O}\bigl{(}\beta^{n}\bigr{)}\to
0\qquad{as}\quad n\to\infty$ (2.13)

uniformly in $x\in\mathbb{K}$. Similarly to the case $R_{n}(s;x)$, taking into
account (2.13) we find the following representation:

$\frac{u^{n}(x)}{\Delta_{n}(x)}=\frac{1}{h(x)-1}+\frac{v(x)\bigl{[}1-u^{n}(x)\bigr{]}}{1-u(x)}+\sum_{k=1}^{n}\varepsilon_{k}(x)u^{k}(x),$
(2.14)

where
$\sup_{x\in\mathbb{K}}\bigl{|}\varepsilon_{n}(x)\bigr{|}\leq\varepsilon_{n}\to{0}$
as $n\to\infty$.

In our further discussion we will also need expansions functions $h(x)$ and
$u(x)$ in the left neighborhood of the point $x=1$.

######  Lemma 1.

Let $\beta<1$ and $\alpha<\infty$. Then for GF $h(x)=\textsf{E}x^{V}$ the
following local expansion holds:

$1-h(x)\sim\frac{1}{1-\beta}(1-x)-\frac{2\beta(1-\beta)+b_{q}}{2(1-\beta)^{3}}(1-x)^{2}\qquad{as}\quad
x\uparrow 1,$ (2.15)

where $b_{q}:=f_{q}^{\prime\prime}(1-)$.

###### Proof.

We write the Peano’s form Taylor expansion for $h(x)=\textsf{E}x^{V}$:

$h(x)=1+h^{\prime}(1-)(x-1)+\frac{h^{\prime\prime}(1-)}{2}(x-1)^{2}+o(x-1)^{2}\qquad{as}\quad
x\uparrow 1.$ (2.16)

Formula (2.5) and standard calculations produce that

$h^{\prime}(1-)=\frac{1}{1-\beta}\qquad{and}\qquad
h^{\prime\prime}(1-)=\frac{2\beta(1-\beta)+b_{q}}{(1-\beta)^{3}}\raise
1.5pt\hbox{.}$

Substituting these expressions in the expansion (2.16), entails (2.15).

The lemma is proved. ∎

Similar arguments can be used to verify the validity of the following lemma.

######  Lemma 2.

Let $\beta<1$ and $\alpha<\infty$. Then

$u(x)=\beta x\bigl{[}1-\gamma_{q}(1-x)\bigr{]}+{\rho}(x),$ (2.17)

where

${\frac{{\rho}(x)}{(1-x)^{2}}}\to{const}\qquad{as}\quad x\uparrow 1.$

###### Proof.

Write the Taylor expansion with Lagrange error bound for $f^{\prime}_{q}(y)$:

$f^{\prime}_{q}(y)=\beta+f^{\prime\prime}_{q}(1)(y-1)+r(y),$

where $r(y)\leq{A}\cdot(y-1)^{2}$ as $y\uparrow{1}$ and $A=const$. Since
$u(x)=xf_{q}^{\prime}\bigl{(}h(x)\bigr{)}$, taking herein $y=h(x)$ and using
(2.15) leads to (2.17).

The lemma is proved. ∎

The following two results directly follow from Lemma 1 and Lemma 2
respectively.

######  Lemma 3.

Let $\beta<1$, $\alpha<\infty$. Then

$h\bigl{(}e^{\theta}\bigr{)}-1\sim\frac{\theta}{1-\beta}+\frac{2+\beta\gamma_{q}}{2(1-\beta)^{2}}\theta^{2}\qquad{as}\quad\theta\to{0}.$
(2.18)

######  Lemma 4.

Let $\beta<1$, $\alpha<\infty$. Then

$\frac{u\bigl{(}e^{\theta}\bigr{)}}{\beta}-1=(1+\gamma_{q})\theta+{{\rho}(\theta)},$
(2.19)

where ${{\rho}(\theta)}={\mathcal{O}^{\ast}}\left({\theta}^{2}\right)$ as
$\theta\to{0}$.

Next Lemma follows from combination of (2.14), (2.18) and (2.19).

######  Lemma 5.

Let $\beta<1$, $\alpha<\infty$. Then

$\frac{\Delta_{n}\bigl{(}e^{\theta}\bigr{)}}{u^{n}(e^{\theta}\bigr{)}}=\frac{1}{1-\beta}\theta+{\mathcal{O}^{\ast}}\bigl{(}{\theta^{2}}\bigr{)}\qquad{as}\quad\theta\to{0}$
(2.20)

for any fixed $n\in\mathbb{N}$.

Now we prove the following lemma.

######  Lemma 6.

Let $\beta<1$, $\alpha<\infty$. Then

$\ln\prod_{k=0}^{n-1}u_{k}\bigl{(}e^{\theta}\bigr{)}\sim-\left(1-\frac{u\bigl{(}e^{\theta}\bigr{)}}{\beta}\right)n-\gamma_{q}\theta\cdot\sum_{k=0}^{n-1}u^{k}\bigl{(}e^{\theta}\bigr{)}\qquad{as}\quad\theta\to{0}$
(2.21)

for any fixed $n\in\mathbb{N}$.

###### Proof.

Using the inequality $\ln(1-y)\geq-y-y^{2}\bigl{/}(1-y)$, which is valid for
$0\leq y<1$, we have

$\displaystyle\ln\prod_{k=0}^{n-1}u_{k}\bigl{(}e^{\theta}\bigr{)}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\ln\left\\{1-\left[1-u_{k}\bigl{(}e^{\theta}\bigr{)}\right]\right\\}$
(2.22) $\displaystyle=$
$\displaystyle\sum_{k=0}^{n-1}\left[u_{k}\bigl{(}e^{\theta}\bigr{)}-1\right]+\rho_{n}^{(1)}(\theta)=:I_{n}(\theta)+\rho_{n}^{(1)}(\theta),$

where

$I_{n}(\theta)=-\sum_{k=0}^{n-1}\left[1-u_{k}\bigl{(}e^{\theta}\bigr{)}\right],$
(2.23)

and

$-\sum\limits_{k=0}^{n-1}\frac{\left[1-u_{k}\bigl{(}e^{\theta}\bigr{)}\right]^{2}}{u_{k}\bigl{(}e^{\theta}\bigr{)}}\leq\rho_{n}^{(1)}(\theta)\leq
0.$

It is easy to see that the sequence of functions $\left\\{h_{k}(x)\right\\}$
does not decrease in $k\in\mathbb{N}$. Then, by the property of the GF, and
the function $u_{k}\bigl{(}e^{\theta}\bigr{)}$ is non-decreasing in $k$, for
any fixed $n\in\mathbb{N}$ and $\theta\in\mathbb{R}$. Therefore,

$\frac{1-u_{0}\bigl{(}e^{\theta}\bigr{)}}{u_{0}\bigl{(}e^{\theta}\bigr{)}}I_{n}(\theta)\leq\rho_{n}^{(1)}(\theta)\leq
0.$ (2.24)

According to the GF property, we will also verify that under our conditions
$1-u_{0}\bigl{(}e^{\theta}\bigr{)}\to 0$ as $\theta\to 0$. Then, according to
(2.24), $\rho_{n}^{(1)}(\theta)\to 0$ if only $I_{n}(\theta)$ has a finite
limit as $\theta\to 0$.

Using the Taylor formula, we write

$f_{q}^{\prime}(t)=f_{q}^{\prime}(t_{0})-f_{q}^{\prime\prime}(t_{0})(t_{0}-t)+(t_{0}-t)g(t_{0};t),$

where $g(t_{0};t)=(t_{0}-t)f_{q}^{\prime\prime\prime}(\tau)\bigl{/}2$ and
$t_{0}<\tau<t$. Hence, at $t_{0}=h(x)$ and $t=h_{k}(x)$ we have the following
relation:

$u_{k}(x)=\frac{u(x)}{\beta}-\frac{xf_{q}^{\prime\prime}\left(h(x)\right)}{\beta}\Delta_{k}(x)+\Delta_{k}(x)g_{k}(x),$

where $g_{k}(x)=x\Delta_{k}(x)f_{q}^{\prime\prime\prime}(\tau)\bigl{/}2\beta$
and $h_{k}(x)<\tau<h(x)$. Therefore,

$u_{k}\bigl{(}e^{\theta}\bigr{)}=\frac{u\bigl{(}e^{\theta}\bigr{)}}{\beta}-\frac{e^{\theta}f_{q}^{\prime\prime}\left(h\bigl{(}e^{\theta}\bigr{)}\right)}{\beta}\Delta_{k}\bigl{(}e^{\theta}\bigr{)}+\Delta_{k}\bigl{(}e^{\theta}\bigr{)}g_{k}\bigl{(}e^{\theta}\bigr{)}.$

Then (2.23) becomes

$I_{n}(\theta)=-\left[1-\frac{u\bigl{(}e^{\theta}\bigr{)}}{\beta}\right]n-\frac{e^{\theta}f_{q}^{\prime\prime}\left(h\bigl{(}e^{\theta}\bigr{)}\right)}{\beta}\sum_{k=0}^{n-1}\Delta_{k}\bigl{(}e^{\theta}\bigr{)}+\rho_{n}^{(2)}(\theta),$
(2.25)

where

$0\leq\rho_{n}^{(2)}(\theta)\leq\Delta_{0}\bigl{(}e^{\theta}\bigr{)}\sum_{k=0}^{n-1}g_{k}\bigl{(}e^{\theta}\bigr{)}.$

In the last step we used the fact that
$\bigl{|}\Delta_{n}(x)\bigr{|}\leq\beta^{n}\bigl{|}\Delta_{0}(x)\bigr{|}$
which follows from inequality (2.12). It follows from (2.18) that
$\Delta_{0}\bigl{(}e^{\theta}\bigr{)}=\mathcal{O}(\theta)$ as $\theta\to{0}$.
And also the asymptotic estimation (2.13) implies that
$g_{k}\bigl{(}e^{\theta}\bigr{)}=\mathcal{O}\bigl{(}\beta^{k}\bigr{)}$ as
$k\to\infty$ and hence the functional series
$\sum_{k=0}^{\infty}g_{k}\bigl{(}e^{\theta}\bigr{)}$ converges for all
$\theta\in\mathbb{R}$. Therefore,

$\Delta_{0}\bigl{(}e^{\theta}\bigr{)}\sum_{k=0}^{n-1}g_{k}\bigl{(}e^{\theta}\bigr{)}=\mathcal{O}(\theta)\to{0}\qquad{as}\quad\theta\to{0}.$

Then the remainder term in (2.25)

$\rho_{n}^{(2)}(\theta)\to{0}\qquad{as}\quad\theta\to{0}.$ (2.26)

Assertion (2.20) implies that

$\sum_{k=0}^{n-1}\Delta_{k}\bigl{(}e^{\theta}\bigr{)}=\frac{\theta}{1-\beta}\sum_{k=0}^{n-1}u^{k}\bigl{(}e^{\theta}\bigr{)}\left(1+\mathcal{O}^{\ast}\bigl{(}{\theta}\bigr{)}\right)\qquad{as}\quad\theta\to{0}.$
(2.27)

Since
${e^{\theta}f^{\prime\prime}_{q}\left(h\bigl{(}e^{\theta}\bigr{)}\right)}\to{f^{\prime\prime}_{q}(1)}$
as $\theta\to{0}$, combining relations (2.22), (2.25)–(2.27) and, after some
calculations, we will come to (2.21).

The lemma is proved. ∎

##  3\. Proof of Theorems

###### Proof of Theorem 1.

Define a sequence of variables

${\zeta_{n}}:=\frac{S_{n}-\textsf{E}S_{n}}{\mathcal{K}_{n}}$

for some positive real-valued sequence $\mathcal{K}_{n}$ such that
$\mathcal{K}_{n}\to\infty$ as $n\to\infty$ and then an appropriate
characteristic function

${\varphi_{\zeta_{n}}(\theta)}:=\textsf{E}\left[\exp\bigl{\\{}i\theta\zeta_{n}\bigr{\\}}\right]=\textsf{E}\left[\theta_{n}^{S_{n}}\cdot\exp\left\\{\frac{-i\theta\textsf{E}S_{n}}{\mathcal{K}_{n}}\right\\}\right],$

where $\theta_{n}:=\exp\left\\{i\theta\big{/}\mathcal{K}_{n}\right\\}$ and
$\theta\in\mathbb{R}$. Using (2.4) we write

$\ln{\varphi_{\zeta_{n}}(\theta)}\sim-\left(1+\gamma_{q}\right)\frac{i\theta}{\mathcal{K}_{n}}n+\ln{T_{n}\left(\theta_{n}\right)}\qquad{as}\quad
n\to\infty,$ (3.1)

where $T_{n}(x)=\textsf{E}x^{S_{n}}$. Simultaneously according to (2.11) and
Lemma 6,

$\ln{T_{n}\left(\theta_{n}\right)}\sim-\left(1-\frac{u\left(\theta_{n}\right)}{\beta}\right)n-\frac{i\theta\gamma_{q}}{\mathcal{K}_{n}}\cdot\sum_{k=0}^{n-1}u^{k}\left(\theta_{n}\right)$
(3.2)

as $n\to\infty$. In turn, (2.19) implies

$-\left(1-\frac{u\left(\theta_{n}\right)}{\beta}\right)n=\left(1+\gamma_{q}\right){\frac{i\theta}{\mathcal{K}_{n}}}n+n\rho\left({\frac{i\theta}{\mathcal{K}_{n}}}\right),$
(3.3)

where
$0<\lim_{\theta\to{0}}{\rho(\theta)\big{/}{\theta}^{2}}=:{C_{\rho}}<\infty$.
Now we readily choose

$\mathcal{K}_{n}=\mathcal{O}^{\ast}\left(\sqrt{n}\right)\qquad{as}\quad
n\to\infty$ (3.4)

which is equivalent to $\mathcal{K}_{n}\big{/}\sqrt{n}\to{C_{\mathcal{K}}}>0$.
Hence we see that

$n\rho\left({\frac{i\theta}{\mathcal{K}_{n}}}\right)\to-{K}{\theta^{2}}\qquad{as}\quad
n\to\infty,$ (3.5)

where $K:={C_{\rho}}\big{/}{C^{2}_{\mathcal{K}}}>0$. At the same time, since
$u(x)=xf_{q}^{\prime}\bigl{(}h(x)\bigr{)}$, in our assumptions we observe that
$u(x)\leq\beta$ uniformly in $x\in[0,1]$. Therefore, one can choose
$\varepsilon>0$ so desirably small that

$\left|u^{k}\left(\theta_{n}\right)-\beta^{k}\right|\leq\varepsilon$

for large enough $n$. This entails that
${\lim_{n\to\infty}}\sum_{k=0}^{n-1}u^{k}\left(\theta_{n}\right)$ converges
uniformly in $\theta\in\mathbb{R}$. Eventually, after combination of
asymptotic estimations (3.2)–(3.5), and denoting $\sigma^{2}:=2C_{\rho}$, the
relation (3.1) becomes

$\ln{\varphi_{\zeta_{n}}(\theta)}=-{\frac{\sigma^{2}\theta^{2}}{2}}+{\mathcal{K}_{n}(\theta)},$
(3.6)

where
${\mathcal{K}_{n}(\theta)}=\mathcal{O}^{\ast}\left({i\theta}\big{/}{\mathcal{K}_{n}}\right)$
as $n\to\infty$. Finally, we conclude that

${\varphi_{\zeta_{n}}(\theta)}\longrightarrow\exp\left\\{-\frac{\sigma^{2}\theta^{2}}{2}\right\\}\qquad{as}\quad
n\to\infty$

for any fixed $\theta\in\mathbb{R}$. The assertion follows now from the
continuity theorem for characteristic functions.

Theorem 1 is proved. ∎

###### Proof of Theorem 2.

The relation (3.6) and formal use of inequalities

$\left|e^{iy}\right|\leq{1}\qquad{and}\qquad\bigl{|}e^{iy}-1-y\bigr{|}\leq{\frac{|y|^{2}}{2}}$

imply

$\displaystyle\left|{\varphi_{\zeta_{n}}(\theta)}-{e^{-{\sigma^{2}\theta^{2}}/{2}}}\right|$
$\displaystyle\leq$
$\displaystyle\left|{e^{-{\sigma^{2}\theta^{2}}/{2}}}\right|\left|e^{\mathcal{K}_{n}(\theta)}-1\right|$
(3.7) $\displaystyle\leq$
$\displaystyle\left|e^{\mathcal{K}_{n}(\theta)}-1-{\mathcal{K}_{n}(\theta)}\right|+\bigl{|}{\mathcal{K}_{n}(\theta)}\bigr{|}$
$\displaystyle\leq$
$\displaystyle\frac{\bigl{[}{\mathcal{K}_{n}(\theta)}\bigr{]}^{2}}{2}+\bigl{|}{\mathcal{K}_{n}(\theta)}\bigr{|}$

for all $n$. By definition we write

${\mathcal{K}_{n}(\theta)}={C(n)}\frac{i\theta}{\mathcal{K}_{n}}\raise
1.5pt\hbox{,}$

where $\lim_{n\to{\infty}}{C(n)}=C<\infty$. Then, denoting

$F_{n}(x):=\textsf{P}\bigl{\\{}{\zeta_{n}}<x\bigr{\\}},$

and using the estimation (3.7), we obtain the Berry-Esseen approximation bound
[3, p. 538] as follows:

$\displaystyle\Bigl{|}F_{n}(x)-{\Phi}_{0,\sigma^{2}}(x)\Bigr{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\pi}\int\limits_{-T}^{T}\left|\frac{{\varphi_{\zeta_{n}}(\theta)}-{e^{-{\sigma^{2}\theta^{2}}/{2}}}}{\theta}\right|d\theta+\frac{24{M}}{\pi{T}}$
(3.8) $\displaystyle\leq$
$\displaystyle\frac{2}{\pi}{\frac{C(n)}{\mathcal{K}_{n}}}T+\frac{24{M}}{\pi{T}}$

for all $x$ and $T>0$, where ${M}$ is such that
$\Phi^{\prime}_{0,\sigma^{2}}(x)\leq{M}$. It can be decidedly taken that
${M}={1}\bigl{/}{\sigma\sqrt{2\pi}}$.

We let $T\to\infty$ and in the same time it is necessary to be
$T=o\left(\sqrt{n}\right)$ since
$\mathcal{K}_{n}=\mathcal{O}^{\ast}\left(\sqrt{n}\right)$. We can choose $T$
in general, in the form of $T=n^{\delta}{\mathcal{L}_{T}(n)}$, where
$0<\delta<1/2$ and $\mathcal{L}_{T}(n)$ slowly varies at infinity in the sense
of Karamata. Then we reform (3.8) as follows:

$\Bigl{|}F_{n}(x)-{\Phi}_{0,\sigma^{2}}(x)\Bigr{|}\leq\frac{\mathcal{L}_{C}(n)}{n^{1/2-\delta}}+\frac{\mathcal{L}_{M}(n)}{n^{\delta}}\raise
1.5pt\hbox{,}$ (3.9)

where

$\mathcal{L}_{C}(n):=\frac{2C(n)}{\pi}{\mathcal{L}_{T}(n)}\qquad{and}\qquad\mathcal{L}_{M}(n):=\frac{24M}{\pi}\frac{1}{\mathcal{L}_{T}(n)}\raise
1.5pt\hbox{.}$

To come up to optimum degree of an estimation of approximation in (3.9), we
would choose value of $\delta$ such that $(1/2-\delta)\delta$ has reached the
maximum value for $\delta\in(0,1/2)$. It happens only in a unique case when
$\delta=1/2-\delta$ or $\delta=1/4$. Thus (3.9) becomes

$\Bigl{|}F_{n}(x)-{\Phi}_{0,\sigma^{2}}(x)\Bigr{|}\leq\frac{\mathcal{L}(n)}{n^{1/4}}\raise
1.5pt\hbox{,}$

where ${\mathcal{L}(n)}={\mathcal{L}_{C}(n)}+{\mathcal{L}_{M}(n)}$ slowly
varies at infinity.

The theorem proof is completed. ∎

###### Proof of Theorem 3.

First we will show that

$\frac{S_{n}}{n}\buildrel\textsf{P}\over{\longrightarrow}{1+\gamma_{q}}\qquad{as}\quad
n\to\infty.$ (3.10)

Writing

$\eta_{n}:=\frac{S_{n}}{n}=\frac{\textsf{E}S_{n}}{n}+\frac{\mathcal{K}_{n}}{n}{\zeta_{n}},$

and considering (2.4), we have

$\displaystyle{\varphi_{\eta_{n}}(\theta)}$ $\displaystyle:=$
$\displaystyle\textsf{E}\left[\exp\bigl{\\{}i\theta\eta_{n}\bigr{\\}}\right]$
(3.11) $\displaystyle=$ $\displaystyle
e^{i\theta\left(1+\gamma_{q}\right)}\bigl{[}{\varphi_{\zeta_{n}}(\theta)}\bigr{]}^{{\mathcal{K}_{n}}/n}\left(1-{\frac{i\theta\gamma_{q}}{1-\beta}}\frac{1}{n}{\bigl{(}1-\beta^{n}\bigr{)}}\right),$

where
${\varphi_{\zeta_{n}}(\theta)}=\textsf{E}\left[\exp\bigl{\\{}i\theta\zeta_{n}\bigr{\\}}\right]$.
Relation (3.6) implies

${\varphi_{\zeta_{n}}(\theta)}={e^{-{\sigma^{2}\theta^{2}}/{2}}}\left(1+\mathcal{O}^{\ast}\left({i\theta}\big{/}{\mathcal{K}_{n}}\right)\right)\qquad{as}\quad
n\to\infty$

and hence
$\bigl{[}{\varphi_{\zeta_{n}}(\theta)}\bigr{]}^{{\mathcal{K}_{n}}/n}\to{0}$ as
$n\to\infty$. Thus (3.11) entails

${\varphi_{\eta_{n}}(\theta)}\to{e^{i\theta\left(1+\gamma_{q}\right)}}\qquad{as}\quad
n\to\infty.$

According to the continuity theorem, this is sufficient for being of (3.10).

From (3.11) we obtain

$\displaystyle\left|{\varphi_{\eta_{n}}(\theta)}-{e^{i\theta\left(1+\gamma_{q}\right)}}\right|$
$\displaystyle\leq$
$\displaystyle\left|\bigl{[}{\varphi_{\zeta_{n}}(\theta)}\bigr{]}^{{\mathcal{K}_{n}}/n}\left(1-{\frac{i\theta\gamma_{q}}{1-\beta}}\frac{1}{n}{\bigl{(}1-\beta^{n}\bigr{)}}\right)-1\right|$
$\displaystyle\leq$
$\displaystyle\left|{\frac{i\theta\gamma_{q}}{1-\beta}}\frac{1}{n}{\bigl{(}1-\beta^{n}\bigr{)}}\right|.$

We accounted in the last step that $|{\varphi_{\ast}(\theta)}|\leq 1$ for any
characteristic function. Now we can write the Berry–Esseen bound as follows:

$\displaystyle\Bigl{|}\textsf{P}\bigl{\\{}\eta_{n}<x\bigr{\\}}-I_{1+\gamma_{q}}(x)\Bigr{|}$
$\displaystyle\leq$
$\displaystyle\frac{1}{\pi}\int\limits_{-T}^{T}\left|\frac{{\varphi_{\eta_{n}}(\theta)}-{e^{i\theta\left(1+\gamma_{q}\right)}}}{\theta}\right|d\theta+\frac{24{M}_{\eta}}{\pi{T}}$
$\displaystyle\leq$
$\displaystyle\frac{\gamma_{q}}{\pi}{\frac{\bigl{(}1-\beta^{n}\bigr{)}}{1-\beta}}\frac{2T}{n}+\frac{24}{\pi{T}}\leq\frac{1}{\pi}{\frac{2\gamma_{q}}{1-\beta}}\frac{T}{n}+\frac{24}{\pi{T}}\raise
1.5pt\hbox{,}$

where we put ${M}_{\eta}=1$ which is suitable for the degenerate distribution
function.

In this case we choose $T=n^{\delta}{\mathcal{L}_{T}(n)}$, where $0<\delta<1$
and $\mathcal{L}_{T}(n)$ slowly varies at infinity. Therefore

$\Bigl{|}\textsf{P}\bigl{\\{}\eta_{n}<x\bigr{\\}}-I_{1+\gamma_{q}}(x)\Bigr{|}\leq\frac{\mathcal{L}_{\beta}(n)}{n^{1-\delta}}+\frac{\mathcal{L}_{1}(n)}{n^{\delta}}\raise
1.5pt\hbox{,}$ (3.12)

where

$\mathcal{L}_{\beta}(n):=\frac{1}{\pi}{\frac{2\gamma_{q}}{1-\beta}}{\mathcal{L}_{T}(n)}\qquad{and}\qquad\mathcal{L}_{1}(n):=\frac{24}{\pi}\frac{1}{\mathcal{L}_{T}(n)}\raise
1.5pt\hbox{.}$

We find $\delta=1/2$ and (3.12) becomes

$\Bigl{|}\textsf{P}\bigl{\\{}\eta_{n}<x\bigr{\\}}-I_{1+\gamma_{q}}(x)\Bigr{|}\leq\frac{\mathcal{L}_{\gamma}(n)}{n^{1/2}}\raise
1.5pt\hbox{,}$

where
${\mathcal{L}_{\gamma}(n)}={\mathcal{L}_{\beta}(n)}+{\mathcal{L}_{1}(n)}$
slowly varies at infinity.

The proof is completed. ∎

## References

  * [1] Asmussen S. and Hering H. Branching processes. Birkhäuser, Boston, 1983.
  * [2] Athreya K. B. and Ney P. E. Branching processes. Springer, New York, 1972.
  * [3] Feller W. An Introduction to Probability Theory and its Applications. vol.1. John Wiley & Sons, 1968.
  * [4] Harris T. E. The theory of branching processes. Springer-Verlag, Berlin, 1963.
  * [5] Imomov A. A. On Markov continuous time analogue of Q-processes. Theory Prob. and Math. Stat., 2012, 84, 57–64.
  * [6] Imomov A. A. Limit Theorem for the Joint Distribution in the Q-processes. Journal of Siberian Federal University. Mathematics and Physics, 2014. v. 7(3), 289–296.
  * [7] Jagers P. Branching Progresses with Biological applications. JW & Sons, Pitman Press, GB, 1975.
  * [8] Karpenko A. V. and Nagaev S. V. Limit theorems for the total number of descendents for the Galton-Watson branching process. Theory Probab. Appl., 1994, 38, 433–455.
  * [9] Kennedy D. P. The Galton-Watson process conditioned on the total progeny. Jour. Appl. Prob., 1975, 12, 800–806.
  * [10] Kolchin V. F. Random mappings. Nauka, Moscow, 1984. (Russian)
  * [11] Pakes A. G. Some limit theorems for the total progeny of a branching process. Adv. App. Prob., 1971, 3(1), 176–192.
  * [12] Sevastyanov B. A. Branching processes. Nauka, Moscow, 1971. (Russian)


# A Distance-Deviation Consistency and Model-Independent Method to Test the
Cosmic Distance-Duality Relation

Chichun zhou  Jian Hu<EMAIL_ADDRESS>Maocai Li  Xi Zhang  School of
Engineering, Dali University, Dali 671003, China  Guanwen Fang
<EMAIL_ADDRESS>School of Mathematics and Physics, Anqing Normal
University, Anqing 246011, China

(Accepted for publication in The Astrophysical Journal)

###### Abstract

A distance-deviation consistency and model-independent method to test the
cosmic distance duality relation (CDDR) is provided. The method is worth
attention on two aspects: firstly, a distance-deviation consistency method is
used to pair subsamples: instead of pairing subsamples with redshift deviation
smaller than a value, say $\left|\Delta z\right|<0.005$. The redshift
deviation between subsamples decreases with the redshift to ensure the
distance deviation stays the same. The method selects more subsamples at high
redshift, up to $z=2.16$, and provides 120 subsample pairs. Secondly, the
model-independent method involves the latest data set of $1048$ type Ia
supernovae (SNe Ia) and $205$ strong gravitational lensing systems (SGLS),
which are used to obtain the luminosity distances $D_{L}$ and the ratio of
angular diameter distance $D_{A}$ respectively. With the model-independent
method, parameters of the CDDR, the SNe Ia light-curve, and the SGLS are
fitted simultaneously. The result shows that $\eta=0.047^{+0.190}_{-0.151}$
and CDDR is validated at 1$\sigma$ confidence level for the form
$\frac{{{D_{L}}}}{{{D_{A}}}}{(1+z)^{-2}}=1+\eta z$.

Cosmology,cosmic distance duality, SGL

##  1 Introduction

The cosmic distance duality relation (CDDR), also called Etherington’s
reciprocity relation (Etherington, 1933), plays an important role in modern
cosmology, especially in galaxy observations (Cunha, Marassi & Shevchuk, 2007;
Mantz et al., 2010), cosmic microwave background (CMB) radiation observations
(Komatsu et al., 2011), and the gravitational lensing (Ellis, 2007). The CDDR
reads

$\frac{{{D_{L}}}}{{{D_{A}}}}{(1+z)^{-2}}=1,$ (1)

where $D_{L}$ is the luminosity distance, $D_{A}$ is the angular diameter
distance, and $z$ is the redshift. The CDDR is valid for all cosmological
models based on Riemannian geometry. The basis of this relation is that the
number of photons is conservative and photons travel along the null geodesic
in a Riemannian space-time (Ellis, 2007).

The validity of the CDDR is explored widely for the past decades, because any
deviation of CDDR may trigger new physics. Uzan et al. (2004) investigates the
possible deviation from the CDDR by analysing the measurements of SZE and
X-ray emission data of galaxy clusters and reports that the parameter
$\eta=0.89^{+0.04}_{-0.03}$ and is at 1$\sigma$ confidence level. Holanda,
Lima & Ribeiro (2011) takes more parametrized forms of $\eta$ and found no
departure from the CDDR. Nair et al. (2011); Basset & Kunz (2004); Holanda et
al. (2010); Holanda, Lima & Ribeiro (2011); Cao & Liang (2011); Meng et al.
(2012)used $D_{L}$ directly from type Ia supernovae (SNe Ia) to test the CDDR.
Hu, & Wang (2018) and Melia (2018) used the $D_{L}$ form SNe Ia with the
$R_{h}=ct$ cosmology model to test the CDDR and compared the model with the
$\Lambda$CDM model. Basset & Kunz (2004) found a 2$\sigma$ violation of the
CDDR using the luminosity distances $D_{L}$ from SNe Ia and the angular
diameter distance $D_{A}$ from FRIIb radio galaxies. Räsänen et al. (2016)
used CMB anisotropy to test the CDDR. The CDDR is also important in studying
cosmic opacity (Lv & Xia, 2016; Hu et al., 2017).

To test the CDDR, many $D_{L}$ and $D_{A}$ pairs at the same redshift $z$ need
to be provided simultaneously. In principle, the two distances should neither
be correlated nor based on any cosmology models. That is, a model-independent
method and a quality and quantity collection of sample pairs are important.
Conventionally, in determining $D_{L}$, the method of Standard Candles (e.g.
SNe Ia, GRB (Wang & Dai, 2006; Wang et al., 2007; Wang & Dai, 2008; Wang et
al., 2009; Wang & Dai, 2006; Wang et al., 2015; Tu & Wang, 2018; Wang & Wang,
2019)) plays a prominent part. However, the method of Standard Candles is
model-dependent, i.e., a special cosmology model is used in calibrating the
light-curve parameters. For example, Suzuki et al. (2012) used the cold dark
matter (CDM), wCDM, and owCDM models to fit the parameters of Union2.1 SNe Ia
and to constrain the cosmology parameters. In determining $D_{A}$, the method
of using the Sunyaev–Zel’dovich effect (SZE) and X-ray observations (Cavaliere
& Fusco-Fermiano, 1978; Sunyaev & Zel’dovich, 1972; Bonamente et al., 2006) is
important in finding $D_{A}$ from galaxy clusters. $D_{A}$ can also be
obtained from ultra-compact radio sources (Li & Lin, 2018) and baryon acoustic
oscillations (BAOs) (Wu et al., 2015).

There are model-independent methods. Liao et al. (2016) introduced a new
method to test the CDDR based on strong gravitational lensing systems and SNe
Ia. In their work, they constrain $\eta$, the parameter of the SNe Ia light-
curve, and the parameter of the SGLS simultaneously. Räsänen et al. (2016)
uses the temperature-redshift relation of CMB to test the CDDR, in their work,
a flat FRW universe is assumed. Ruan et al. (2018) use a similar model-
independent method with the SGLS, the SNe Ia, and the HII galaxy Hubble
diagram to test the CDDR. To avoid the effect of the cosmic opacity, Liao
(2019) uses the $D_{L}$ from the gravitational wave signals and the ratio of
$D_{A}$ from the SGLS (for details see below) with a model-independent method,
which is proposed in (Liao et al., 2016). These model independent methods show
that the CDDR is valid in the given redshift range, say, $z<1.0$.

In this paper, we provide a distance-deviation consistency and model-
independent method to test the CDDR. The distance-deviation consistency method
pairs subsamples with redshift deviation decrease with the redshift to ensure
the distance deviation stays the same. It is because the distance grows
nonlinearly with the redshift, the larger the redshift is, the smaller the
redshift deviation is between two sources with the same distance deviation.
The latest data set of SNe Ia with $1048$ samples and strong gravitational
lensing system (SGLS) with $205$ samples are involved and the distance-
deviation consistency method enables us to take full advantage of the data:
the method selects more subsamples at high redshift, up to $z=2.16$, and
provide 121 subsample pairs. With the model-independent method, parameters of
the CDDR, the SNe Ia light-curve, and the SGLS are fitted simultaneously.

This paper is organized as follows. In Sec.2, we introduce the latest data set
of the SNe Ia with $1048$ samples and the SGLS with $205$ samples. In Sec.3,
we explain a distance-deviation consistency method to pair subsamples and
describe the method of statistical analysis. The numerical results are shown.
Conclusions and discussions are given in Sec.4.

##  2 Data

In this section, we describe two sets of data suitable for testing the CDDR,
one based on the redshift and the light-curve of the SNe Ia, from which we can
obtain $D_{L}(z)$, the other base on the observational velocity dispersion of
the lens galaxy, redshifts of the strong gravitational lensing system (SGLS),
from which we can obtain the ratio of $D_{A}(z)$.

###  2.1 The Pantheon SNe Ia Sample

In this section, we introduce the contents of the Pantheon sample (Scolnic et
al., 2018). The Pantheon sample consisting of a total of 1048 SNe Ia in the
range of $0.01<z<2.3$ is constructed by a subset include 279 SNe Ia
($0.03<z<0.68$) from the Pan-STARRS1 (PS1) Medium Deep Survey, the Sloan
Digital Sky Survey (SDSS), the SNLS, the various low-z, and Hubble Space
Telescope (HST) samples.

On one hand, the luminosity distances $D_{L}(z)$ can be determined accurately
by multiple light-curve fitters (e.g., (Jha et al., 2007; Guy et al., 2010;
Burns et al., 2011; Mandel et al., 2011)). From the phenomenological point of
view, the distance modulus, $\mu$, of an SNe Ia can be extracted from its
light curve. In a modified version of the Tripp formula (Tripp et al., 1998),
the SALT2 light-curve fit parameters are transformed into distances:

$\mu=m_{B}-M+\alpha x_{1}-\beta c+\Delta_{M}+\Delta_{B},$ (2)

where $m_{B}$ is the apparent magnitude, $M$ is the absolute B-band magnitude
of a fiducial SNe Ia with $x_{1}=0$ and $c=0$, $\Delta_{M}$ and $\Delta_{B}$
are distance corrections based on the mass of the host galaxy of the SNe Ia
and predicted biases from simulations respectively. $\alpha$ and $\beta$ are
light-curve parameters of relations between the luminosity and the stretch and
between the luminosity and the color respectively. Moreover, $\Delta_{M}$ in
the equation (2) can be written in the form Scolnic et al. (2018)

$\Delta_{M}=\gamma[1+e^{(-(m-m_{step})/\tau)}]^{-1},$ (3)

where $\gamma$, $m_{step}$, and $\tau$ are coefficients to be determined
(Scolnic et al., 2018). On the other hand, one assumes that the SNe Ia with
identical color, shape, and galactic environment have on average the same
intrinsic luminosity for all redshifts (Betoule et al., 2014). According to
the definition of the distance modulus, it can be written as

$\mu=5\log(\frac{D_{L}}{\rm Mpc})+25.$ (4)

By using equations (2) and (4), we can obtain the luminosity distances
$D_{L}(z)$ for the Pantheon SNe Ia Sample.

###  2.2 The strong gravitational lensing system Sample

For the new SGLS sample, we used from Amante et al. (2019), which contains 205
SGLS. This sample constituted from some survey projects, (e.g. the SLACS, the
CASTLES survey, the BELLS, the LSD …). In an SGLS, the light is bent by
massive bodies (e.g., galaxy, galaxy cluster) which is predicted by the
general theory of relativity. The SGLS is a powerful astrophysical tool to
explore the universe and galaxy and has been rapidly developed in recent
years, especially like dark energy (Biesiada, 2006; Biesiada et al., 2010; Cao
et al., 2012, 2015; Jullo et al., 2010; Magaña et al., 2015, 2018), the CDDR
(Liao et al., 2016; Liao, 2019), the cosmic acceleration (Tu et al, 2019),
calibrating the standard candles (Wen et al., 2019), and cosmological models’
comparison(Melia et al., 2015; Leaf & Melia, 2018; Tu et al, 2019). In an
SGLS, a single galaxy acting as the lens, the Einstein radius depends on three
parameters: depends on the angular distance to the source and between the lens
and the source, and the mass distribution within the lensing galaxy. A
singular isothermal sphere (SIS) model (Ratnatunga et al, 1999) is used to
describe the lens galaxy’s mass distribution. The ratio of the angular
diameter distances between lens and source and between observer and source can
be obtained from a special physical model (e.g. SIS model). Because these
distances depend on the cosmological metric, the ratio can be used to
constrain cosmological parameters.

In an SIS model of the SGLS, the distance ratio $R^{A}(z_{l},z_{s})$
($D^{A}_{l}s/D^{A}_{s}$) is related to observables in the following way
(Biesiada et al., 2010),

$R^{A}(z_{l},z_{s})=\frac{c^{2}\theta_{E}}{4\pi\sigma^{2}_{SIS}},$ (5)

where $c$ is the speed of light, $\theta_{E}$ is the Einstein radius, and
$\sigma_{SIS}$ is the velocity dispersion of the stellar in the periphery of
the lens galaxy due to the lens mass distribution in the SIS model. In
general, $\sigma_{SIS}$ not equals to the observed stellar velocity dispersion
$\sigma_{0}$ (White & Davis, 1996). To express the difference, researchers use
a phenomenological free parameter $f_{e}$ defined by the relation
$\sigma_{SIS}=f_{e}\sigma_{0}$ (Kochanek, 1992; Ofek et al, 2003; Cao et al.,
2012), where $(0.8)^{1/2}<f_{e}<(1.2)^{1/2}$. In this case, the systematic
error is caused by $\sigma_{0}$ as $\sigma_{SIS}$, the deviation of the SIS
model, the effects of secondary lenses (nearby galaxies), the line of sight
contamination (Ofek et al, 2003), etc. The uncertainty of equation (5) can be
written by (Liao et al., 2016)

$\sigma_{R^{A}}(z_{l},z_{s})=R^{A}(z_{l},z_{s})\sqrt{(4\delta_{\sigma_{sis}})^{2}+(\delta_{\theta_{E}})^{2}}.$
(6)

In equation(6), $\delta_{\sigma_{sis}}$ and $\delta_{\theta_{E}}$ are the
fractional uncertainty of the $\sigma_{sis}$ and $\theta_{E}$, respectively.
To test the CDDR, the left term of equation (ref5), $R^{A}(z_{l},z_{s})$,
should be expressed as the ratio of luminosity distance,
$D^{A}_{ls}/D^{A}_{s}$. We transform $R^{A}(z_{l},z_{s})$ into the ratio of
Comoving distance($D_{C}$) or dimensionless distance $H_{0}D_{C}/c$. In a flat
space, the dimensionless distance satisfies

$d(z_{l},z_{s})=d(z_{s})-d(z_{l}).$ (7)

By using the equation,

$D_{A}(z)=\frac{D_{C}(z)}{1+z}=\frac{H_{0}d(z)}{c(1+z)},$ (8)

$R^{A}(z_{l},z_{s})$ can be written as

$R^{A}(z_{l},z_{s})=1-\frac{(1+z_{l})D_{A}(z_{l})}{(1+z_{s})D_{A}(z_{s})}.$
(9)

In a non-flat space, the expression of $R^{A}(z_{l},z_{s})$ is more
complicated, one can refer to Räsänen et al. (2015). Fortunately, most
cosmological tests today support a flat cosmic (Planck Collaboration et al.,
2016), in this work, we test the CDD relation in the case where we assume that
space-time is flat.

To test the CDDR, equation (1) is rewritten by the parametrization of the
deviation:

$\frac{{{D_{L}}}}{{{D_{A}}}}{(1+z)^{-2}}=1+\eta z.$ (10)

Combining equations (9) and (10), $R^{A}(z_{l},z_{s})$ can be written as

$R^{A}_{0}(z_{l},z_{s})=1-\frac{(1+\eta z_{s})(1+z_{s})d_{L}(z_{l})}{(1+\eta
z_{l})(1+z_{l})d_{L}(z_{s})}.$ (11)

By using equation (4), the part $\frac{d_{L}(z_{l})}{d_{L}(z_{s})}$ of
equation (11) can be rewritten as

$\begin{split}\lg[\frac{d_{L}(z_{l})}{d_{L}(z_{s})}]=&0.2\\{m_{B}(z_{l})-m_{B}(z_{s})+\alpha[x(z_{l})-x(z_{s})]-\\\
&\beta[c(z_{l})-c(z_{s})]+\Delta_{M}(z_{l})-\Delta_{M}(z_{s})\\},\end{split}$
(12)

where, the absolute magnitude of SNe Ia subsample is offset, $z_{l}$ is the
redshift of the lens, and $z_{s}$ is the redshift of the sources.

##  3 Method and results

In this section, we introduce a distance-deviation consistency method of data
selection and show the result.

###  3.1 Method of data selection: a distance-deviation consistency method

To test the CDDR, $D_{L}$ and $D_{A}$ at the same redshift $z$ need to be
provided simultaneously. However, redshifts of subsamples from the SGLS and
the SNe Ia are different. To take full advantage of the data, one needs to
pair the subsamples efficiently. In this section, we provide a distance-
deviation consistency method to pair subsamples, which outperforms the
conventional method.

The redshift-difference of subsample-pairs in this work is not fixed, and it
decreases with the redshifts to ensure the distance deviation of the sources
stays the same. The relation between $\Delta z$ and coordinate distance with a
cosmology model reads

$\Delta{d_{c}^{model}(z)}=d_{c}^{model}(z+\Delta z^{model})-d_{c}^{model}(z),$
(13)

where $R_{h}=ct$ and flat $\Lambda$CDM model with $\Omega_{m}=0.31$(Planck
Collaboration et al., 2016) are used. By setting $\Delta d_{c}/d_{c}$ equals
5% and combining equation(13), $\Delta{z}(z)$ can be calculated:

$\Delta{z}(z)=min\\{\Delta{z}^{\Lambda CDM}(z),\Delta{z}^{R_{h}=ct}(z)\\},$
(14)

where the distance formulas of the two cosmology model, $\Lambda$CDM and
$R_{h}=ct$ are used. The $R_{h}=ct$ cosmological model was proposed in Melia
(2007). In the $R_{h}=ct$ universe, the luminosity distance $D_{L}$ can be
written by:

$D_{L}^{R_{h}=ct}(z)=\frac{c}{H_{0}}(1+z)\ln(1+z)$ (15)

This model is a Friedmann–Robertson–Walker (FRW) cosmological model, which is
obeying the cosmological principle and Weyl’s postulate(Melia, 2007; Melia &
Shevchuk, 2012). In $R_{h}=ct$ universe that space expands at a constant rate,
rather than an accelerating rate. Theoretically, there are some controversies
with this model, which focus on the zero active mass condition
$\rho$$+3p=0$(for more details see Melia (2016); Kim, Lasenbyet & Hobson
(2016); Melia (2017)). In terms of the fitting of observational data, the
model performs relatively well. The originator of this model himself and his
collaborators have done extensive work comparing it with the standard model
using many different types of observations, and they have found that the
$R_{h}=ct$ model is better than the Standard Model(e.g.Melia (2013);
Melia&Manoj (2018); Fatuzzo&Melia (2017b); Melia (2014); Wei et al. (2015);
Melia (2019)). Additional work by others also illustrates this point(Yu &
Wang, 2014; Yuan & Wang, 2015). There is also a lot of work against this
model(e.g. Tutusaus et al. (2016); Shafer (2015)) Thus, despite the
theoretical controversy, this model has gained some support for the data, and
we can use this model jointly with the standard model to select the data.

For a given SGLS subsample, the redshift-dimensionless distance relation,
equation(14), is used to acquire a suitable redshift deviation interval. Then,
the subsamples of the SNe Ia with redshifts within the interval are selected
as the candidate. Finally, the subsample with the smallest redshift-deviation
is selected.

Our method outperforms the conventional method in two aspects: (1) information
of subsamples at high-redshifts is conserved. To pair subsamples with a slight
difference of redshift is a simple and commonly used method. For example,
$\Delta z=0.005$ (Holanda et al., 2010, 2012; Holanda, Busti, & Alcaniz, 2016;
Li et al, 2011; Nair et al., 2011), $\Delta z=0.006$ (Goncalves, Holanda&
Alcaniz, 2012), and $\Delta z=0.003$ (Liao, 2019). Gaussian Process (GP)
reconstruction is also a usable method (Zhang, 2014; Ruan et al., 2018). The
linear interpolation method is also taken by some researchers (Liang et al.,
2013; Hu, & Wang, 2018). These methods reduce the systematic error to an
extent but do not consider the distance-deviation consistency. The relation
between the distance and the redshift is non-linear: the same redshift
deviation at the higher redshift has a smaller distance deviation. For
example, some researchers set the $\Delta z=0.005$, the uncertainty of
dimensionless distance is 5% at $z\sim 0.1$, the minimum redshift of the
subsample we select, but small than 1% at $z\sim 1$ for selecting data with a
general cosmology model, so the selecting uncertainty is not the consistency
of distance-deviation. The selecting uncertainty of their methods are all
ignoring, and if they were taking them into account in their fitting, they had
to introduce a cosmological model so that their methods were no longer model-
independent. In our approach, although two cosmological models are introduced,
they are used to jointly pick the data, breaking the dependence on a single
cosmological model when picking the data, and are not introduced into the
$\chi^{2}$ function, in other words, the final parameter fit results are
independent of the cosmological model. Cao et al. (2017)used a similar method
for selecting the data with a $\Lambda$CDM model to fitting the parameters of
the ultra-compact radio quasars. At high redshift, in general, subsamples are
sparse and matching pairs of subsamples is difficult if a fixed $\Delta z$ is
used. The distance deviation consistency method selects more subsamples at
high-redshifts.

(2) More subsample pairs are selected. For example, in the work of Liao et al.
(2016), they gain only 60 pairs of samples to testing the CDDR. According to
figure(1), at z=0.11, the most previously used $\Delta z=0.005$ and the curve
of $\Delta d/d$ in this method are intersecting. So, the $\Delta d/d=5\%$ is
the maximum allowable uncertainty of dimensionless distance. In other words,
if the error of selecting exceeds 5%, the statistical error of our results
must be higher than that of previous work. If the error is significantly lower
than 5%, the number of data pairs we choose will not be significantly
improved. With the method of fixed $\Delta d/d=5\%$, the number of the pairs
can reach 68. The data utilization increases by 13.3%. In this work, we have
obtained 120 pairs of samples that redshift from 0.11 to 2.16.

A comparison between our method with a fixed $\Delta d/d$ and the conventional
method with a fixed $\Delta z$ are shown in figures (1) and (2). In figure
(1), the deviation, $\Delta d/d$, falls rapidly with the increase of redshift
with $\Delta z$ fixed. The main advantage of a fixed $\Delta d/d$ is shown in
figure (2). To conclude, on the one hand, our method maintains more
information about subsamples at high-redshifts. On the other hand, our method
selects more subsample-pairs thus reduces the systematic error.

Figure 1: The relation between the $\Delta d/d$ and $z$,the red line and the
blue line indicate the relationship between the relative error of $d$ and the
redshift in $R_{h}=ct$ model and $\Lambda CDM$ model, respectively. The black
line indicates the scheme we used.
Figure 2: The relation between the $\Delta_{z}$ and $z$

###  3.2 Method of statistical analysis

To determine the parameters, we minimize $\chi^{2}$ function. By using
equations (5) and (11), $\chi^{2}$ function can be written as

$\chi^{2}=\sum^{120}_{1}(\frac{(R^{A}(z_{l},z_{s})-R^{A}_{0}(z_{l},z_{s}))^{2}}{\sigma^{2}_{R^{A}(z_{l},z_{s})}+\sigma^{2}_{R^{A}_{0}(z_{l},z_{s})}+\sigma^{2}_{sel}}),$
(16)

where $\sigma_{R^{A}(z_{l},z_{s})}$ is the uncertainty of the SGLS with SIS
model, $\sigma_{R^{A}_{0}(z_{l},z_{s})}$ is the error caused by the
uncertainty of the distance modulus of SNe Ia, which is related to the
uncertainty of observed data (e.g. $m_{B}$), and $\sigma_{sel}$ is the
uncertainty of data selection, which is related to artificial selection. By
using equation(11), the uncertainty of data selection can be written as

$\sigma_{sel}=\frac{(1+\eta z_{s})(1+z_{s})d_{L}(z_{l})}{(1+\eta
z_{l})(1+z_{l})d_{L}(z_{s})}*\sqrt{(\frac{\Delta
d_{l}}{d_{l}})^{2}+(\frac{\Delta d_{s}}{d_{s}})^{2}},$ (17)

where $\frac{\Delta d_{l}}{d_{l}}=\frac{\Delta d_{s}}{d_{s}}=\frac{\Delta
d}{d}=5\%$.

We use the Markov chainMonte Carlo (MCMC) method to constrain the parameters
in equation(17). EMCEE (Foreman-Mackey et al., 2013) python package is used.
In order to execute the MCMC process, we need to provide the prior values
first. In Scolnic et al. (2018), the best value and the 1$\sigma$ confidence
level of the parameters are shown. But in our work, we take an SIS model of
SGLS to calibrating these parameters, which will different. So, we take the
prior interval that completely covering the value range of the value from
Scolnic et al. (2018). The prior probability for parameters
$P(\alpha,\beta,f_{e},\eta,\gamma,m_{step},\tau)$ is the product of prior
probability of each parameter. The prior probability is assumed to be uniform
distributions: $P(\alpha)=U[-0.2,0.2]$, $P(\beta)=U[2,6]$,
$P(f_{e})=U[0.5,1.5]$, $P(\eta)=U[-0.5,1.5]$, $P(\gamma)=U[0,0.3]$,
$P(M_{step})=U[5,15]$, $P(\tau)=U[0.001,1]$. In Pantheon samples, the errors
include both statistic and systematic deviation. The systematic error
relatives to all data point and appears as a huge covariance matrix. In this
work, part of SNe Ia are selected, only the statistic error is considered.

###  3.3 Results

The result is shown in figure (3) and table (1). Triangle contours are plotted
by using the open-source python package “Getdist”. One can see from figure (3)
that the best-fitted center value is $\eta=0.047^{+0.190}_{-0.151}$, which is
at 1$\sigma$ confidence level. The result indicates that the CDDR is in
agreement with the observations and there are no signs of violation in light
of SN Ia and SL data.

Figure 3: The 2D regionsand 1D marginalized distributions with 1$\sigma$ and
2$\sigma$ contours for the parameters $\alpha$, $\beta$, $f_{e}$, $\gamma$,
$m_{step}$,$\tau$ and $\eta$ using Pantheon sample and SGLS sample. Table 1:
Constraints on the coefficients of light-curve parameters and $\eta$ at the
1$\sigma$ confidence levels.

parameters | value
---|---
$\alpha$ | $0.001^{+0.061}_{-0.061}$
$\beta$ | $5.283^{+0.417}_{-0.464}$
$f_{e}$ | $1.046^{+0.020}_{-0.019}$
$\eta$ | $0.047^{+0.190}_{-0.151}$
$\gamma$ | $0.141^{+0.080}_{-0.070}$
$m_{step}$ | $10.055^{+0.177}_{-0.148}$
$\tau$ | $0.134^{+0.048}_{-0.073}$
$\chi^{2}$ | 117.430
$\chi^{2}/{d.o.f}$ | 117.430/113

##  4 DISCUSSIONS AND CONCLUSIONS

The validation of the CDDR is a crucial topic in cosmology. Any violation of
the CDDR may generate a new theory of physics. In recent years, to compare the
$D_{L}$ derived from SNe la and the $D_{A}$ measured using galaxy clusters is
the common method to test the CDDR. To use this method, a specific cosmology
model with some parameters (e.g. the matter density parameter $\Omega_{m}$,
the cosmic equation of state, and the Hubble constant) must be assumed. Such
results are hardly convincing.

In testing the CDDR, using a model-independent method is necessary. Moreover,
to obtain a data sample contains a large number of $D_{L}$ and $D_{A}$ pairs
are also important. However, the number of useful subsample pairs is limited
by the observed data, and pair subsamples with redshift-deviation smaller than
a constant will lose the subsamples at high redshift which leads to systematic
errors.

In this paper, we provide a distance-deviation consistency and a model-
independent method to test the CDDR. By applying the distance-deviation
consistency method on the latest data set of SNe Ia with $1048$ samples and
strong gravitational lensing system (SGLS) with $205$ samples, we obtain the
collection of subsample pairs not only contains more subsamples but also
maintains the information of subsamples at high redshift, up to $z=2.16$. By
applying a model-independent method: the SGLS model is used to replace the
cosmology model in SNe Ia light-curve fitting, the result shows that
$\eta=0.047^{+0.190}_{-0.151}$ and CDDR is validated at 1$\sigma$ confidence
level for the form $\frac{{{D_{L}}}}{{{D_{A}}}}{(1+z)^{-2}}=1+\eta z$.

We thank the anonymous referee for constructive comments.This work is
supported by the Research Foundation for Advanced Talents(NO
KY1916102940,KY1916102740), Dali university and the Special Foundation for
Basic Research Plan (grants KY2013113740 and KY2013114440)

## References

  * Amante et al. (2019) Amante, M. H., et al. 2019, arXiv:1906.04107
  * Basset & Kunz (2004) Basset, B. A., Kunz, M., 2004, PRD, 69, 101305
  * Betoule et al. (2014) Betoule, M., et al. 2014, A&A, 568, A22
  * Biesiada (2006) Biesiada, M., 2006, PRD, 73, 023006
  * Biesiada et al. (2010) Biesiada, M., et al. 2010, MNRAS, 406, 1055
  * Bonamente et al. (2006) Bonamente, M., et al. 2006, ApJ, 647, 25
  * Burns et al. (2011) Burns, C. R., Stritzinger, M., Phillips, M. M., et al. 2011, AJ, 141, 19
  * Cao et al. (2012) Cao, S., et al., 2012, JCAP, 03, 016
  * Cao et al. (2015) Cao, S., Biesiada, M., Gavazzi, R., Piorkowska, A., Zhu, Z. U., 2015, ApJ, 806, 185
  * Cao et al. (2017) Cao, S., et al. 2017, JCAP, 02, 012
  * Cao & Liang (2011) Cao, S., Liang, N., 2011, RAA, 11, 10
  * Cavaliere & Fusco-Fermiano (1978) Cavaliere, A., Fusco-Fermiano, R., 1978, A&A., 667, 70
  * Cunha, Marassi & Shevchuk (2007) Cunha, J. V., Marassi, L., Lima, J. A. S., 2007, MNRAS, 379, L1
  * Ellis (2007) Ellis, G. F. R., 2007, Gen. Relativ. Gravit, 39, 1047
  * Etherington (1933) Etherington, I. M. H., 1933, Philos. Mag, 15, 761
  * Fatuzzo&Melia (2017b) Fatuzzo, M., &Melia F. 2017, ApJ, 846, 2
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., Goodman, J., 2013, PASP, 125, 306
  * Goncalves, Holanda& Alcaniz (2012) Goncalves, R. S., Holanda, R. F. L., Alcaniz, J. S., 2012, MNRAS, 420, L43
  * Guy et al. (2010) Guy, J., Sullivan, M., Conley, A., et al. 2010, A&A, 523, A7
  * Holanda, Busti, & Alcaniz (2016) Holanda, R. F. L., Busti, V.C., Alcaniz, J. S., 2016, JCAP, 02, 054
  * Holanda et al. (2010) Holanda, R. F. L., Lima, J. A. S., Ribeiro, M. B., 2010, ApJL, 722, L233
  * Holanda, Lima & Ribeiro (2011) Holanda, R. F. L., Lima, J. A. S., Ribeiro, M. B., 2011, A&A, 528, L14
  * Holanda et al. (2012) Holanda, R. F. L., Lima, J. A. S., Ribeiro, M. B., 2012, A&A, 538, A131
  * Hu et al. (2017) Hu, J., Yu, H., Wang, F. Y., 2017, ApJ, 836, 1
  * Hu, & Wang (2018) Hu, J., Wang, F. Y., 2018, MNRAS, 477, 5064
  * Jha et al. (2007) Jha, S., Riess, A. G., Kirshner, R. P., 2007, ApJ, 659, 122
  * Jullo et al. (2010) Jullo, E., Natarajan, P., Kneib, J.-P., et al. 2010, Sci, 329, 924
  * Kim, Lasenbyet & Hobson (2016) Kim, D. Y., Lasenby, A. N., Hobson, M. P., 2016, MNRAS, 460, L119
  * Kochanek (1992) Kochanek, C. S., 1992, ApJ, 384, 1
  * Komatsu et al. (2011) Komatsu, E., et al. 2011, ApJS, 192, 2
  * Leaf & Melia (2018) Leaf, K., Melia, F., 2018, MNRAS, 478, 5104
  * Li & Lin (2018) Li, Xin, & Lin, H. N., 2018, MNRAS, 474, 313
  * Li et al (2011) Li, Z. X., Wu, P. X., Yu, H. W. 2011, ApJL, 729, L14
  * Liang et al. (2013) Liang, N., Li, Z. X., Wu, P. X., Cao, S., Liao, K., Zhu, Z. H. 2013, MNRAS, 436, 1017
  * Liao et al. (2016) Liao, K., et al. 2016, ApJ, 822, 2
  * Liao (2019) Liao, K. 2019, arXiv:1906.09588
  * Lv & Xia (2016) Lv, M. Z., Xia J. Q. 2016, PDU, 13, 139
  * Magaña et al. (2015) Magaña, J. Motta, V., Cárdenas, V. H., Verdugo, T., Jullo E. 2015, ApJ, 813, 69
  * Magaña et al. (2018) Magaña, J., et al. 2018, ApJ, 865, 122
  * Mandel et al. (2011) Mandel, K. S., Narayan, G., & Kirshner, R. P. 2011, ApJ, 731, 120
  * Mantz et al. (2010) Mantz, A. et al. 2010, MNRAS, 406, 1759
  * Melia (2007) Melia, F. 2007, MNRAS, 382, 1917
  * Melia & Shevchuk (2012) Melia, F., Shevchuk, A. S. H. 2012, MNRAS, 419, 2579
  * Melia (2013) Melia, F. 2013, ApJ, 764, 6
  * Melia (2014) Melia, F. 2014, ApJ, 149, 72
  * Melia et al. (2015) Melia, F., Wei, J.J., Wu, X. F. 2015, ApJ, 149, 2
  * Melia (2016) Melia, F. 2016a, FP, 11, 119801
  * Melia (2017) Melia, F. 2017, FP, 12, 129802
  * Melia (2018) Melia, F. 2018, MNRAS, 481, 4855
  * Melia&Manoj (2018) Melia, F.,& Yennapureddy, Manoj K, 2018, MNRAS,480,2
  * Melia (2019) Melia, F. 2019, MNRAS, 489, 517
  * Meng et al. (2012) Meng, X. L., Zhang, T. J., Zhan, H., Wang, X. 2012, ApJ, 745, 98
  * Nair et al. (2011) Nair, R., Jhingan, S., Jain D. 2011, JCAP, 05, 023
  * Ofek et al (2003) Ofek, E. O., Rix, H.-W., & Maoz, D. 2003, MNRAS, 343, 639
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R. et al. 2016, A&A, 594, A13
  * Räsänen et al. (2015) Räsänen, S., Bolejko, K., & Finoguenov, A. 2015, PRL, 115, 101301
  * Räsänen et al. (2016) Räsänen, S., Valiviita J., Kosonen, V. 2016, JCAP, 04, 050
  * Ratnatunga et al (1999) Ratnatunga, K., U.,et al. 1999, AJ, 117, 2010
  * Ringermacher & Mead (2016) Ringermacher, H. I., Mead, L. R. 2016, arXiv:1611.00999
  * Ruan et al. (2018) Ruan, C. Z., Melia, F., Zhang, T. J. 2018, ApJ, 866, 31
  * Scolnic et al. (2018) Scolnic, et al. 2018, ApJ, 859, 101
  * Shafer (2015) Shafer, D. L. 2015, PRD, 91, 103516
  * Sunyaev & Zel’dovich (1972) Sunyaev, R. A., Zel’dovich Y. B. 1972, Comments Astrophys. Space Phys., 4, 173
  * Suzuki et al. (2012) Suzuki, N., et al. 2012, ApJ, 746, 58
  * Tripp et al. (1998) Tripp, R. 1998, A&A, 331, 815
  * Tu & Wang (2018) Tu, Z. L., Wang, F. Y. 2018, ApJ, 869, 2
  * Tu et al (2019) Tu, Z. L., Hu J., Wang, F. Y. 2019, MNRAS, 484, 4337
  * Tutusaus et al. (2016) Tutusaus, I., et al. 2016, PRD, 94, 103511
  * Uzan et al. (2004) Uzan, J. P., Aghanim, N., Mellier, Y. 2004, PRD, 70, 083533
  * Wang & Dai (2006) Wang, F. Y., Dai, Z. G. 2006, MNRAS, 368, 371
  * Wang et al. (2007) Wang, F. Y., Dai, Z. G., Zhu, Z. H. 2007, ApJ, 667, 1
  * Wang & Dai (2008) Wang, F. Y., Dai, Z. G. 2006, MNRAS, 368, 371
  * Wang et al. (2009) Wang, F. Y., Dai, Z. G., Qi, S. 2009, A&A, 507, 1
  * Wang & Dai (2006) Wang, F. Y., Dai, Z. G. 2009, MNRAS, 400, 1
  * Wang et al. (2015) Wang, F. Y., Dai, Z. G., Liang, E. W. 2015, New Astronomy Reviews, 67, 1
  * Wang & Wang (2019) Wang, Y. Y., Wang F. Y. 2019, ApJ, 873, 1
  * Wei et al. (2015) Wei, J. J., Wu, X. F., Melia, F. 2015, MNRAS, 463, 1114
  * Wen et al. (2019) Wen, X. D., Liao, K., 2019, arXiv:1907.02693
  * White & Davis (1996) White, R. E., Davis, D. S. 1996, BAAS, 28, 1323
  * Wu et al. (2015) Wu, P., Li, Z., Liu, X., & Yu, H. 2015, PRD, 92, 023520
  * Yu & Wang (2014) Yu, H., Wang, F. Y. 2014, EPJC, 74, 3090
  * Yuan & Wang (2015) Yuan, C. C., Wang, F. Y. 2015, MNRAS, 452, 2423
  * Zhang (2014) Zhang, Y. 2019, arXiv:1408.3897