© Wuhan University 2026

0 Introduction

Branching process is a mathematical model used to describe the reproduction of species, which has received much attention. As the reproduction of species is subject to factors such as natural environment and social environment, many mathematicians have improved the classical branching process model. Smith and Wilkison first proposed branching processes in random environments and studied the extinction probability of branching processes in independent and identically distributed (i.i.d.) random environments and some properties of branching processes in Markovian environments^[1-3]. Athreya and Karlin^[4-5] discussed the extinction probability and some limiting behaviors of branching processes in stationary ergodic random environments. Holzheimer^[6] introduced the controlled branching model in a random environment and studied its extinction probability. Yanev et al^[7] investigated the sufficient condition for certain extinction and non-certain extinction of controlled branching processes in i.i.d. random environments; Bi et al^[8] proposed a criterion for certain extinction of controlled branching processes in stationary ergodic random environments. Li et al^[9] studied the Markov property and some limiting behaviors of controlled branching processes in random environments. Branching process with migration in random environments was first introduced by Key^[10]. Li et al^[11] discussed the conditional expectations and Markov property of branching processes with migration in random environments; Ren et al^[12] studied the limiting behaviors of branching processes with migration in random environments. Li^[13] investigated the asymptotic behavior for critical branching processes with migration. Wang et al^[14] studied the limit theory for a supercritical branching process; Ren et al^[15-16] studied the Markov property and limiting behaviors of the branching process affected by viral infectivity in a random environment, and the Markov property, conditional probability generating function and sufficient condition for certain extinction of bisexual branching processes influenced by viral infectivity in random environments. There have been many research results on the branching process in random environments, which can be seen in Refs. [17-21]. On the basis of Refs [12,15], the Markov property, conditional probability generating function, conditional expectation and sufficient condition for certain extinction of branching processes with migration and affected by viral infectivity in random environments are considered in this paper. Also, the existing correlation theory of branching process in random environments is generalized.

The remainder of this paper is organized as follows. In Section 1, some notations, definitions are introduced. Sections 2-4 are devoted to presenting the main results, including the Markov property, probability generating functions, conditional expectation and extinction probability. Section 5 is the conclusion.

1 Preliminaries

The model we shall be concerned with in this paper may be described as follows.

Let $(\mathrm{\Omega },F,P)$ be a given probability space, $(\mathrm{\Theta },\mathrm{\Sigma })$ be a measurable space, and $\mathbb{N}=\{\mathrm{0,1},\mathrm{2},\cdots \},$${\mathbb{N}}^{+}=\{\mathrm{1,2},\mathrm{3},\cdots \},$

${\mathbb{N}}^m=\{-m,-m+\mathrm{1},\cdots ,\mathrm{0,1},\cdots \},m\in \mathbb{N}$

Let $\stackrel{\rightharpoonup }{\xi }=\{{\xi }_{\mathrm{0}},{\xi }_{\mathrm{1}},{\xi }_{\mathrm{2}},\cdot \cdot \cdot \}$ be a sequence of i.i.d. random variables, mapping from $(\mathrm{\Omega },F,P)$ to $\mathrm{\Theta }$ and Let $\{I_{ni},n\in \mathbb{N},i\in {\mathbb{N}}^{+}\}$ and $\{X_{ni},n\in \mathbb{N},i\in {\mathbb{N}}^{+}\}$ denote two random variable sequences from $(\mathrm{\Omega },F,P)$ to $\mathbb{N}.$

For any $\theta \in \mathrm{\Theta },$ we assume $\{p_k(\theta ),k\in \mathbb{N}\}$ and $\{{\alpha }_k(\theta ),k\in {\mathbb{N}}^m\}$ are two sequences of probability distributions, satisfying ${\displaystyle \sum _{k=\mathrm{0}}^{\infty }}k{p}_k(\theta )<\mathrm{\infty },\mathrm{0}\le p_{\mathrm{0}}(\theta )+p_{\mathrm{1}}(\theta )<\mathrm{1},{\displaystyle \sum _{k=-m}^{\infty }}k{\alpha }_k(\theta )$

$<\mathrm{\infty },\mathrm{0}\le {\alpha }_{\mathrm{0}}(\theta )<\mathrm{1}$ and there exists $k\in \{-\mathrm{1},-\mathrm{2},\cdot \cdot \cdot ,-m\}$ such that ${\alpha }_k(\theta )>\mathrm{0}.$

Definition 1   Suppose $\{Z_n,n\in \mathbb{N}\}$ and $\{Y_n,n\in \mathbb{N}\}$ are two sequences of random variables mapping from $(\mathrm{\Omega },F,P)$ to $\mathbb{N}$ and $(\mathrm{\Omega },F,P)$ to ${\mathbb{N}}^m$, respectively and satisfy

(i) $Z_{\mathrm{0}}=\mathrm{1},Z_{n+\mathrm{1}}=\{\begin{array}{cc}{\displaystyle \sum _{i=\mathrm{1}}^{Z_n+Y_n}}{X}_{ni}{I}_{ni},& Z_n+Y_n>\mathrm{0}\\ \mathrm{0},& Z_n+Y_n\le \mathrm{0}\end{array},n\in \mathbb{N}$;

(ii) $\begin{array}{l}P(X_{ni}=k|\stackrel{\rightharpoonup }{\xi })=p_k({\xi }_n),k,n\in \mathbb{N},i\in {\mathbb{N}}^{+},\\ P(Y_n=k|\stackrel{\rightharpoonup }{\xi })={\alpha }_k({\xi }_n),n\in \mathbb{N},k\in {\mathbb{N}}^m,\\ P(I_{ni}=x|\stackrel{\rightharpoonup }{\xi })=b^x({\xi }_n)(\mathrm{1}-b({\xi }_n{))}^{\mathrm{1}-x},n\in \mathbb{N},i\in {\mathbb{N}}^{+},x=\mathrm{0,1};\end{array}$

(iii) For given $\stackrel{\rightharpoonup }{\xi }$, $\{X_{ni},n\in \mathbb{N},i\in {\mathbb{N}}^{+}\}$ and $\{I_{ni},n\in \mathbb{N},i\in {\mathbb{N}}^{+}\}$

are both independent sequences; $\{X_{ni},n\in \mathbb{N},i\in {\mathbb{N}}^{+}\}$ and

$\{I_{ni},n\in \mathbb{N},i\in {\mathbb{N}}^{+}\}$ are mutually independent. For fixed $n$, $\{(X_{ni},I_{ni}),i\in {\mathbb{N}}^{+}\}$ are identically distributed. For fixed $n$, $\{Y_n,n\in \mathbb{N}\}$ and $\{Z_n,n\in \mathbb{N}\},\{(X_{ni},I_{ni}),i\in {\mathbb{N}}^{+}\},\{X_{ni},n\in \mathbb{N},$

$i\in {\mathbb{N}}^{+}\}$ are independent respectively.

Then $\{Z_n,n\in \mathbb{N}\}$ is called a branching process with migration and affected by viral infectivity in random environment $\stackrel{\rightharpoonup }{\xi }.$

The above $Z_n,Y_n$ represents the total number of individuals in n-th generation and the number of individuals migrated in n-th generation, respectively; $X_{ni}$ represents the number of individuals in (n+1)-th generation produced by the i-th individual in n-th generation. We suppose $I_{ni}=\mathrm{0}$ when the i-th individual in n-th generation was infected and not cured, that is, it produces no offspring; $I_{ni}=\mathrm{1}$ when the i-th individual in n-th generation didn't have the virus or was cured of it, that is, it can reproduce normally.

2 Markov Property

Since the branching process is a special case of Markov chain in random environments, the branching process in random environments can be studied with the help of the theory of Markov chain. Therefore, we first discuss the Markov property of branching processes with migration and affected by viral infectivity in random environments.

Definition 2   Suppose $\stackrel{\rightharpoonup }{X}=\{X_n,n\in \mathbb{N}\}$ and $\stackrel{\rightharpoonup }{\xi }=\{{\xi }_n,n\in \mathbb{N}\}$ are random sequences from $(\mathrm{\Omega },F,P)$ to $\mathbb{N}$ and $\mathrm{\Theta }$, respectively. For any $x,n\in \mathbb{N}$, if

$P(X_{\mathrm{0}}=x_{\mathrm{0}}^{}|\stackrel{\rightharpoonup }{\xi })=P(X_{\mathrm{0}}=x_{\mathrm{0}}|{\xi }_{\mathrm{0}}),$(1)

$P(X_{n+\mathrm{1}}=x|{\vec{X}}_{\mathrm{0}}^n,\stackrel{\rightharpoonup }{\xi })=P({\xi }_n,X_n,x),$(2)

then $\stackrel{\rightharpoonup }{X}$ is called a Markov chain in random environment $\stackrel{\rightharpoonup }{\xi }$. For ease of exposition, we introduce some notations. Set $\begin{array}{l}A=\{(r_l,x_l):{\displaystyle \sum _{l=\mathrm{1}}^{i+j}}{r}_l{x}_l=k,r_l\in \mathbb{N};x_l=\mathrm{0,1};l=\mathrm{1,2},\cdots ,\\ i+j,i\in {\mathbb{N}}^m,j\in \mathbb{N}\},F_n(\stackrel{\rightharpoonup }{\xi })=\sigma (Z_{\mathrm{0}},Z_{\mathrm{1}},\cdots ,Z_n;\stackrel{\rightharpoonup }{\xi }),n\in \mathbb{N},\\ \begin{array}{l}{P}^{\mathrm{*}i}(\theta ;j,k)=P(Z_{n+\mathrm{1}}=k|Y_n=i,Z_n=j,\theta )\\ =\sum _{(r_l,x_l)\in A}P(X_{n\mathrm{1}}{I}_{n\mathrm{1}}=x_{\mathrm{1}}{r}_{\mathrm{1}},\cdot \cdot \cdot ,X_{n(i+j)}{I}_{n(i+j)}=x_{i+j}{r}_{i+j}|Y_n=i,Z_n=j,\theta )\\ =\{\begin{array}{cc}\sum _{(r_l,x_l)\in A}{\displaystyle \prod _{l=\mathrm{1}}^{i+j}}{p}_{r_l}(\theta )b^{x_l}(\theta )(\mathrm{1}-b{(\theta ))}^{\mathrm{1}-x_l},& i+j>\mathrm{0};\\ {\delta }_{\mathrm{0}k},& i+j\le \mathrm{0}.\end{array}\end{array}\end{array}$

Theorem 1   $\{Z_n,n\in \mathbb{N}\}$ is a Markov chain in random environment $\stackrel{\rightharpoonup }{\xi }$ with the one-step transition probabilities $P({\xi }_n;j,k)={\displaystyle \sum _{i=-m}^{\infty }}{P}^{*i}({\xi }_n;j,k){\alpha }_i({\xi }_n).$

Proof   By Definition 2, to prove $\{Z_n,n\in \mathbb{N}\}$ is a Markov chain in random environment $\stackrel{\rightharpoonup }{\xi }$, it suffices to prove that (1) and (2) hold.

By $Z_{\mathrm{0}}=\mathrm{1}$ we have (1) holds. Next we prove (2) and it suffices to show, for any $k,n\in \mathbb{N}$, it follows that

$P(Z_{n+\mathrm{1}}=k|F_n(\stackrel{\rightharpoonup }{\xi }))=P({\xi }_n;Z_n,k).$(3)

Since $P(Z_{n+\mathrm{1}}=k|F_n(\stackrel{\rightharpoonup }{\xi }))$ and $P({\xi }_n;Z_n,k)$ are measurable with respect to $F_n(\stackrel{\rightharpoonup }{\xi })$, it suffices to show, for any $\mathrm{\Delta }\in F_n(\stackrel{\rightharpoonup }{\xi })$, it follows that

$\underset{\mathrm{\Delta }}{\int }P(Z_{n+\mathrm{1}}=k|F_n(\stackrel{\rightharpoonup }{\xi }))\mathrm{d}P=\underset{\mathrm{\Delta }}{\int }P({\xi }_n;Z_n,k)\mathrm{d}P$(4)

Taking $\mathrm{\Delta }=\{Z_{\mathrm{0}}=k_{\mathrm{0}},\cdots ,Z_{n-\mathrm{1}}=k_{n-\mathrm{1}},Z_n=j;{\xi }_{\mathrm{0}}\in B_{\mathrm{0}},\cdots ,$

${\xi }_l\in B_l\}\in F_n(\stackrel{\rightharpoonup }{\xi })$ (where $k_{\mathrm{0}},k_{\mathrm{1}},\cdots ,k_{n-\mathrm{1}},j\in \mathbb{N};B_{\mathrm{0}},\cdots ,B_l\in \mathrm{\Sigma }$),

we proceed to the proof of (4). We suppose $l\ge n$ (if $l<n$, let $l^{\text{'}}=n$ and $B_{l+\mathrm{1}}=\cdots =B_{l^{\text{'}}}=\mathrm{\Theta }\in \mathrm{\Sigma }$).

Writing $C=\{Z_{\mathrm{0}}=k_{\mathrm{0}},\cdots ,Z_{n-\mathrm{1}}=k_{n-\mathrm{1}},Z_n=j\},D=\{{\xi }_{\mathrm{0}}\in B_{\mathrm{0}},$

$\cdots ,{\xi }_l\in B_l\}$, we get

$\begin{array}{l}\underset{\mathrm{\Delta }}{\int }P({\xi }_n;Z_n,k)\mathrm{d}P=\underset{\mathrm{\Delta }}{\int }P(Z_{n+\mathrm{1}}=k|Z_n,{\xi }_n)\mathrm{d}P\\ =\underset{\mathrm{\Delta }}{\int }{\displaystyle \sum _{i=-m}^{\infty }}P(Z_{n+\mathrm{1}}=k,Y_n=i|Z_n,{\xi }_n)\mathrm{d}P\end{array}$

$\begin{array}{l}=\underset{\mathrm{\Delta }}{\int }{\displaystyle \sum _{i=-m}^{\infty }}P(Z_{n+\mathrm{1}}=k|Y_n=i,Z_n=j,{\xi }_n)P(Y_n=i|Z_n=j,{\xi }_n)\mathrm{d}P\\ =\underset{\mathrm{\Delta }}{\int }{\displaystyle \sum _{i=-m}^{\infty }}P\{{\displaystyle \sum _{l=\mathrm{1}}^{i+j}}{X}_{nl}{I}_{nl}=k|Y_n=i,Z_n=j,{\xi }_n){\alpha }_i({\xi }_n)\mathrm{d}P\\ =\underset{\mathrm{\Delta }}{\int }{\displaystyle \sum _{i=-m}^{\infty }}\sum _{(r_l,x_l)\in A}{\displaystyle \prod _{l=\mathrm{1}}^{i+j}}{p}_{r_l}({\xi }_n)b^{x_l}({\xi }_n)(\mathrm{1}-b({\xi }_n{))}^{\mathrm{1}-x_l}{\alpha }_i({\xi }_n)\mathrm{d}P\\ =\underset{\mathrm{\Delta }}{\int }{\displaystyle \sum _{i=-m}^{\infty }}{P}^{*}({\xi }_n;j,k){\alpha }_i({\xi }_n)\mathrm{d}P.\end{array}$

Using the independence assumed in Definition 1 (iii) and the properties of conditional expectation, it is deduced that

$\begin{array}{l}\underset{\mathrm{\Delta }}{\int }P(Z_{n+\mathrm{1}}=k|F_n(\stackrel{\rightharpoonup }{\xi }))\mathrm{d}P=\underset{\mathrm{\Delta }}{\int }E\{I_{\{Z_{n+\mathrm{1}}=k\}}|F_n(\stackrel{\rightharpoonup }{\xi })\}\mathrm{d}P\\ =\underset{D}{\int }E\{I_{\{Z_{n+\mathrm{1}}=k\}}{I}_C|F_n(\stackrel{\rightharpoonup }{\xi })\}\mathrm{d}P\\ =\underset{\mathrm{\Delta }}{\int }=\underset{D}{\int }{\displaystyle \sum _{i=-m}^{\infty }}E\{I_{\{Z_{n+\mathrm{1}}=k\}}{I}_C|Y_n=i,F_n(\stackrel{\rightharpoonup }{\xi })\}P(Y_n=i|F_n(\stackrel{\rightharpoonup }{\xi }))\mathrm{d}P\\ =\underset{\mathrm{\Delta }}{\int }=\underset{D}{\int }{\displaystyle \sum _{i=-m}^{\infty }}E\{I_{\{{\displaystyle \sum _{l=\mathrm{1}}^{i+j}}{X}_{nl}{I}_{nl}=k\}}{I}_C|Y_n=i,F_n(\stackrel{\rightharpoonup }{\xi })\}{\alpha }_i({\xi }_n)\mathrm{d}P\\ =\underset{D}{\int }{\displaystyle \sum _{i=-m}^{\infty }}P({\displaystyle \sum _{l=\mathrm{1}}^{i+j}}{X}_{nl}{I}_{nl}=k|Y_n=i,F_n(\stackrel{\rightharpoonup }{\xi }))I_C{\alpha }_i({\xi }_n)\mathrm{d}P=\underset{D}{\int }{\displaystyle \sum _{i=-m}^{\infty }}{P}^{*i}({\xi }_n;j,k){\alpha }_i({\xi }_n)\mathrm{d}P.\end{array}$

Therefore, (4) holds, that is, $\{Z_n,n\in \mathbb{N}\}$ is a Markov chain in random environment $\stackrel{\rightharpoonup }{\xi }$ with the one-step transition probabilities claimed above.

3 Conditional Probability Generating Function and Conditional Expectation

Since the extinction probability and martingale convergence of branching process in random environments are often investigated from the conditional probability generating function and conditional expectation of the process, in what follows, we consider the conditional probability generating function and conditional expectation of branching process with migration and affected by viral infectivity in random environments.

For convenience, we give some notations. Writing

$\begin{array}{l}{f}_{{\xi }_n}(s)=E(s^{X_{ni}}|\stackrel{\rightharpoonup }{\xi })={\displaystyle \sum _{k=\mathrm{0}}^{\mathrm{\infty }}}{P}_k({\xi }_n)s^k,\\ g_{{\xi }_n}(s)=E(s^{Y_n}|\stackrel{\rightharpoonup }{\xi })={\displaystyle \sum _{k=-m}^{\infty }}{\alpha }_k({\xi }_n)s^k,\\ {\mathrm{\Pi }}_{{\xi }_n}(s)=E(s^{Z_n}|\stackrel{\rightharpoonup }{\xi })={\displaystyle \sum _{k=\mathrm{0}}^{\mathrm{\infty }}}{P}_{k,n}({\xi }_n)s^k,s\in [\mathrm{0,1}],\\ P_{k,n}({\xi }_n)=P(Z_n=k|\stackrel{\rightharpoonup }{\xi }),k,n\in \mathbb{N}.\end{array}$

Theorem 2   For any $n\in \mathbb{N},s\in [\mathrm{0,1}],$it follows that

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{n+\mathrm{1}}}(s)={\mathrm{\Pi }}_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdot g_{{\xi }_n}(\mathrm{1}-b({\xi }_n)\\ +b({\xi }_n)f_{{\xi }_n}(s))+\hspace{1em}{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{k+r}].\end{array}$(5)

Proof   From Definition 1, the definition of ${\mathrm{\Pi }}_{{\xi }_{n+\mathrm{1}}}(s)$ and properties of conditional expectation, we obtain

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{n+\mathrm{1}}}(s)=E(s^{Z_{n+\mathrm{1}}}|\stackrel{\rightharpoonup }{\xi })={\displaystyle \sum _{k=\mathrm{0}}^{\mathrm{\infty }}}E(s^{{\displaystyle \sum _{i=\mathrm{1}}^{k+Y_n}}{X}_{ni}{I}_{ni}}|Z_n=k,\stackrel{\rightharpoonup }{\xi })P(Z_n=k|\stackrel{\rightharpoonup }{\xi })\\ +{\displaystyle \sum _{k=m}^{\mathrm{\infty }}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{\mathrm{\infty }}}E(s^{{\displaystyle \sum _{i=\mathrm{1}}^{k+r}}{X}_{ni}{I}_{ni}}|Y_n=r,Z_n=k,\stackrel{\rightharpoonup }{\xi }){\alpha }_r({\xi }_n).\\ ={\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n)\{{\displaystyle \sum _{r=-k}^{\mathrm{\infty }}}E(s^{{\displaystyle \sum _{i=\mathrm{1}}^{k+r}}{X}_{ni}{I}_{ni}},Y_n=r|Z_n=k,\stackrel{\rightharpoonup }{\xi })+{\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)\}+{\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}P(Y_n=r|Z_n=k,\stackrel{\rightharpoonup }{\xi })\}\\ +{\displaystyle \sum _{k=m}^{\mathrm{\infty }}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{\mathrm{\infty }}}E(s^{{\displaystyle \sum _{i=\mathrm{1}}^{k+r}}{X}_{ni}{I}_{ni}}|Y_n=r,Z_n=k,\stackrel{\rightharpoonup }{\xi }){\alpha }_r({\xi }_n).\\ +{\displaystyle \sum _{k=m}^{\mathrm{\infty }}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{\mathrm{\infty }}}E(s^{{\displaystyle \sum _{i=\mathrm{1}}^{k+r}}{X}_{ni}{I}_{ni}},Y_n=r|Z_n=k,\stackrel{\rightharpoonup }{\xi })\end{array}$

Since for $n\in \mathbb{N},i\in {\mathbb{N}}^{+},X_{ni},I_{ni}$ are independent, so we have

$\begin{array}{l}E(s^{X_{ni}{I}_{ni}}|\stackrel{\rightharpoonup }{\xi })=E(s^{X_{ni}\cdot \mathrm{0}}|\stackrel{\rightharpoonup }{\xi })P(I_{ni}=\mathrm{0}|\stackrel{\rightharpoonup }{\xi })+E(s^{X_{ni}\cdot \mathrm{1}}|\stackrel{\rightharpoonup }{\xi })P(I_{ni}=\mathrm{1}|\stackrel{\rightharpoonup }{\xi })\\ \hspace{1em}\hspace{1em}\hspace{1em}=\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s).\end{array}$

Hence,

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{n+\mathrm{1}}}(s)={\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n{){\displaystyle \sum _{r=-k}^{\infty }}{\alpha }_r({\xi }_n)[\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s)]}^{k+r}\\ +{\displaystyle \sum _{k=m}^{\mathrm{\infty }}}{P}_{k,n}({\xi }_n{){\displaystyle \sum _{r=-m}^{\infty }}{\alpha }_r({\xi }_n)[\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s)]}^{k+r}\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)\\ ={\mathrm{\Pi }}_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdot g_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{k+r}].\end{array}$

This completes the proof.

Corollary 1   For any $n\in \mathbb{N},s\in [\mathrm{0,1}],$ it follows that

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{\mathrm{1}}}(s)=(\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(s))\cdot g_{{\xi }_{\mathrm{0}}}(\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(s))\\ \hspace{1em}\hspace{1em}+{\displaystyle \sum _{r=-m}^{-\mathrm{2}}}{\alpha }_r({\xi }_{\mathrm{0}})[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{r+\mathrm{1}}]\end{array}$

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{n+\mathrm{1}}}(s)=[\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots )]\\ \cdot {\displaystyle \prod _{i=\mathrm{0}}^n}{g}_{{\xi }_i}(\mathrm{1}-b({\xi }_i)+b({\xi }_i)f_{{\xi }_i}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots ))\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{r+k}]\\ +{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}{\displaystyle \prod _{l=i+\mathrm{1}}^n}{g}_{{\xi }_l}(\mathrm{1}-b({\xi }_l)+b({\xi }_l)f_{{\xi }_l}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots ))\\ \cdot {\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,i}({\xi }_i){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_i)\{\mathrm{1}-[\mathrm{1}-b({\xi }_i)+b({\xi }_i)f_{{\xi }_i}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))\cdots ))]}^{r+k}\}.\end{array}$

Proof   From $P_{\mathrm{1,0}}({\xi }_{\mathrm{0}})=P(Z_{\mathrm{0}}=\mathrm{1}|\stackrel{\rightharpoonup }{\xi })=\mathrm{1}$, we have ${\mathrm{\Pi }}_{{\xi }_{\mathrm{0}}}(s)=s.$ Using the recursion of (5), we obtain

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{\mathrm{1}}}(s)={\mathrm{\Pi }}_{{\xi }_{\mathrm{0}}}(\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(s))\cdot g_{{\xi }_{\mathrm{0}}}(\mathrm{1}-b({\xi }_{\mathrm{0}})\\ +b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(s))+{\displaystyle \sum _{r=-m}^{-\mathrm{2}}}{\alpha }_r({\xi }_{\mathrm{0}})[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{r+\mathrm{1}}]\\ =(\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(s))\cdot g_{{\xi }_{\mathrm{0}}}(\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(s))\\ +{\displaystyle \sum _{r=-m}^{-\mathrm{2}}}{\alpha }_r({\xi }_{\mathrm{0}})[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{r+\mathrm{1}}].\end{array}$

Applying the recursion of Theorem 2, we then have

$\begin{array}{l}{\mathrm{\Pi }}_{{\xi }_{n+\mathrm{1}}}(s)={\mathrm{\Pi }}_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\\ \cdot g_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{k+r}]\\ =g_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\{{\mathrm{\Pi }}_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_{n-\mathrm{1}})\\ +b({\xi }_{n-\mathrm{1}})f_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s)))=[\mathrm{1}-b({\xi }_{\mathrm{0}})+b({\xi }_{\mathrm{0}})f_{{\xi }_{\mathrm{0}}}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots )]\\ \cdot g_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_{n-\mathrm{1}})+b({\xi }_{n-\mathrm{1}})f_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))){\displaystyle \prod _{i=\mathrm{0}}^n}{g}_{{\xi }_i}(\mathrm{1}-b({\xi }_i)+b({\xi }_i)f_{{\xi }_i}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots ))\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n-\mathrm{1}}({\xi }_{n-\mathrm{1}}){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_{n-\mathrm{1}})[\mathrm{1}-(\mathrm{1}-b({\xi }_{n-\mathrm{1}})\\ +b({\xi }_{n-\mathrm{1}})f_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s)))}^{k+r}]\}+{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{k+r}]\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{k+r}]+{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}{\displaystyle \prod _{l=i+\mathrm{1}}^n}{g}_{{\xi }_l}(\mathrm{1}-b({\xi }_l)+b({\xi }_l)f_{{\xi }_l}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots ))\\ =g_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))g_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_{n-\mathrm{1}})+b({\xi }_{n-\mathrm{1}})\\ \cdot f_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s)))\cdot {\mathrm{\Pi }}_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_{n-\mathrm{1}})+b({\xi }_{n-\mathrm{1}})+{\displaystyle \sum _{i=\mathrm{0}}^{n-\mathrm{1}}}{\displaystyle \prod _{l=i+\mathrm{1}}^n}{g}_{{\xi }_l}(\mathrm{1}-b({\xi }_l)+b({\xi }_l)f_{{\xi }_l}(\cdots (\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\cdots ))\\ \cdot f_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s)))+g_{{\xi }_n}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}(s))\\ \cdot {\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n-\mathrm{1}}({\xi }_{n-\mathrm{1}}){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_{n-\mathrm{1}})[\mathrm{1}-(\mathrm{1}-b({\xi }_{n-\mathrm{1}})+b({\xi }_{n-\mathrm{1}})\\ \cdot f_{{\xi }_{n-\mathrm{1}}}(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s)))}^{k+r}]\\ +{\displaystyle \sum _{k=\mathrm{0}}^{m-\mathrm{1}}}{P}_{k,n}({\xi }_n){\displaystyle \sum _{r=-m}^{-k-\mathrm{1}}}{\alpha }_r({\xi }_n)[\mathrm{1}-(\mathrm{1}-b({\xi }_n)+b({\xi }_n)f_{{\xi }_n}{(s))}^{k+r}]\\ =\cdots \end{array}$

This proof is completed.

Theorem 3   For any $n\in \mathbb{N}$, it follows that