Subcritical branching processes in random environment with immigration stopped at zero

TL;DR

This paper analyzes subcritical branching processes with immigration in a random environment, focusing on the tail distribution of their life periods, and proves exponential decay using change of measure and limit theorems.

Contribution

It introduces a detailed analysis of the tail behavior of life periods in subcritical branching processes with immigration in random environments, with new probabilistic techniques.

Findings

Tail distribution decays exponentially

Established limit theorems for associated random walks

Applied change of measure to analyze process behavior

Abstract

We consider subcritical branching processes with immigration which evolve under the influence of a random environment and study the tail distribution of life periods of such processes defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again. We prove that the tail distribution decays with exponential rate. The main tools are the change of measure and some conditional limit theorems for random walks.

Equations

$$F:=(F(\{j\}),j=0,1,...),G:=(G(\{j\}),j=0,1,...)$$

$$\mathcal{L}\left(Y_n|Y_{n-1}=y_{n-1},\mathcal{E}=\{{\mathbf{q}}_1,{\mathbf{q}}_2,...\}\right)=\mathcal{L}({\xi }_{n1}+\dots +{\xi }_{n{y}_{n-1}}+{\eta }_n)$$

$$\mathcal{L}\left(Z_n|Z_{n-1}=z_{n-1},{\mathcal{E}}_1=(f_1,f_2,...)\right)=\mathcal{L}({\xi }_{n1}+\dots +{\xi }_{n{z}_{n-1}}).$$

$$X_n:=\log m\left(F_n\right).$$

$$m(F):=\sum _{j=0}^{\infty }jF(\{j\}).$$

$$W_{n}:=\left\{\begin{array}[]{cc}0,&\text{ if }T_{n}:=\xi_{n1}+\ldots+\xi_{nW_{n-1}}=0,\\[7.22743pt] T_{n}+\eta_{n},&\text{if }T_{n}>0.\end{array}\right.$$

$$\zeta :=\min \left\{n\ge 1:W_n=0\right\}$$

$$F(s):=\sum _{j=0}^{\infty }F\left(\left\{j\right\}\right)s^j,G(s):=\sum _{j=0}^{\infty }G\left(\left\{j\right\}\right)s^j.$$

$$F_n(s):=\sum _{j=0}^{\infty }{F}_n\left(\left\{j\right\}\right)s^j,G_n(s):=\sum _{j=0}^{\infty }{G}_n\left(\left\{j\right\}\right)s^j,s\in [0,1],$$

$$F_{i,n}(s):=$$

$$F_{n,i}(s):=$$

$$N(0;s):=\frac{G_0(s)-G_0(0)}{1-G_0(0)}$$

$$H_n$$

$$H_n^{\ast }$$

$$R_n$$

$$\mathcal{H}(s):=\sum _{n=0}^{\infty }{H}_n{s}^n\text{ , }{\mathcal{H}}^{\ast }(s):=\sum _{n=1}^{\infty }{H}_n^{\ast }{s}^n\text{ and }\mathcal{R}(s):=\sum _{n=1}^{\infty }{R}_n{s}^n.$$

$$\mathcal{R}(s)=\frac{s{\mathcal{H}}^{\ast }(s)+s{R}_1}{1-s\mathcal{H}(s)}.$$

$$-\infty \le \mathbf{E}[X]<0$$

$$\mathbf{E}[X{e}^{\beta X}]=0$$

$$\mathbb{E}[\psi ({\mathbf{Q}}_1,\cdots ,{\mathbf{Q}}_n,W_0,\cdots ,W_n)]:={\gamma }^{-n}\mathbf{E}[\psi ({\mathbf{Q}}_1,\cdots ,{\mathbf{Q}}_n,W_0,\cdots ,W_n)e^{\delta S_n}],$$

$$\gamma :=\mathbf{E}[e^{\delta X}].$$

$$\mathbb{E}[X]=0.$$

$$a_n=n^{1/\alpha }{l}_1(n)$$

$$\vartheta (a):=\sum _{j=a}^{\infty }{j}^{2}F(\{j\})/m(F{)}^2,a\in \mathbb{N}.$$

$$\mathbb{E}\left[({\log }^{+}\vartheta (a){)}^{\alpha +ϵ}\right]<\infty$$

$$\mathbb{E}[{\log }^{+}\vartheta (a)]<\infty$$

$$\mathbb{E}\left[\frac{G_0^{\prime }(1)}{1-G_0(0)}\right]<\infty .$$

$$\mathbf{P}(\zeta >n)\sim \frac{r{\mathcal{H}}^{\ast }(r)+r{R}_1}{\mathcal{H}(r)+r{\mathcal{H}}^{\prime }(r)}{r}^{-n-1};$$

$$\mathbf{P}(\zeta >n)\sim C\frac{{\gamma }^n}{b_n},C\in \left(0,\infty \right)$$

$$\mathbf{P}(\zeta >n)=o({\gamma }^n).$$

$$Y_c(N-1)=0,\underset{N\le k<N+{\zeta }_c}{\min }{Y}_c(k)>0,Y_c(N+{\zeta }_c)=0.$$

$$H_n={\gamma }^{n+1}\mathbb{E}\left[(1-G_0(F_{0,n+1}(0)))\prod _{i=1}^n{G}_i\left(F_{i,n+1}(0)\right)e^{-\delta S_{n+1}}\right]:={\gamma }^{n+1}{\overline{H}}_n,$$

Full text

Keywords: branching processes; random environment; immigration; life period

1   Introduction and statement of main results

Galton-Watson branching processes with immigration are among the popular models of branching processes. Different versions of such processes have found applications in physics, demography, biology and other fields of science.

One of the problems being interesting from theoretical and practical points for such processes is the distribution of the so-called life period of a branching process with immigration defined as the length of the time interval between the moment when first invader (or invaders) came to an empty site until the moment when the site becomes empty again (see, for instance, [4], [11], [14], [16]). Information on the length of such periods may be used, for example, in epidemiology, seismology and ecology. In the contents of epidemics, such periods correspond to the duration of outbreaks of diseases that do not lead to full epidemics and to the period of occupancy of sites in metapopulations [8]. They may be used to analyse the waiting time interval for the end of earthquake aftershocks [9] or plasmid incompatibility [12], [13], and for considering some other models of similar nature.

In this note we consider Galton-Watson branching processes allowing immigration and evolving in a random environment. Individuals in such processes reproduce independently of each other according to offspring distributions which vary in a random manner from one generation to the other. In addition, a number of immigrants join each generation independently of the development of the population and according to the laws varying at random from generation to generation. A formal definition of the process looks as follows. Let $\Delta=\left(\Delta_{1},\Delta_{2}\right)$ be the space of all pairs of probability measures on $\mathbb{N}_{0}=\{0,1,2,\ldots\}.$ Supplying $\Delta$ with the component-wise metric of total variation we obtain a Polish space. Let $\mathbf{Q}=\{F,G\}$ be a two-dimensional random vector with independent components

[TABLE]

taking values in $\Delta$, and let $\mathbf{Q}_{n}=\{F_{n},G_{n}\},n=1,2,\ldots,$ be a sequence of independent copies of $\mathbf{Q}$. The infinite sequence $\mathcal{E}=\left\{\mathbf{Q}_{1},\mathbf{Q}_{2},...\right\}$ is called a random environment.

A sequence of $\mathbb{N}_{0}$-valued random variables $\mathbf{Y}=\left\{Y_{n},\ n\in\mathbb{N}_{0}\right\}$ specified on the respective probability space $(\Omega,\mathcal{F},\mathbf{P})$ is called a branching process with immigration in the random environment (BPIRE), if $Y_{0}$ is independent of $\mathcal{E}$ and, given $\mathcal{E}$ the process $\mathbf{Y}$ is a Markov chain with

[TABLE]

for every $n\in\mathbb{N}:=\mathbb{N}_{0}\backslash\left\{0\right\}$, $y_{n-1}\in\mathbb{N}_{0}$ and $\mathbf{q}_{1}=\left(f_{1},g_{1}\right),\mathbf{q}_{2}=\left(f_{2},g_{2}\right),...\in\mathbf{Q}$, where $\xi_{n1},\xi_{n2},\ldots$ are i.i.d. random variables with distribution $f_{n}$ and independent of the random variable $\eta_{n}$ with distribution $g_{n}$. In the language of branching processes $Y_{n-1}$ is the $(n-1)$th generation size of the population, $f_{n}$ is the distribution of the number of children of an individual at generation $n-1$ and $g_{n}$ is the law of the amount of immigrants joining generation $n$.

Along with the process $\mathbf{Y}$ we consider a branching process $\mathbf{Z}=\left\{Z_{n},\ n\in\mathbb{N}_{0}\right\}$ in the random environment (BPRE) $\mathcal{E}_{1}=\{F_{1},F_{2},...\}$ which, given $\mathcal{E}_{1}$ is a Markov chain with $Z_{0}=1$ and, for $n\in\mathbb{N}$

[TABLE]

An important role in studying BPRE and BPIRE is played by the so-called associated random walk $\mathcal{S}=\left\{S_{0},S_{1},...\right\}$. This random walk has initial state $S_{0}$ and increments $X_{n}=S_{n}-S_{n-1}$, $n\geq 1$, defined as

[TABLE]

Here the increments $X_{n}$ are i.i.d. copies of the logarithmic mean offspring number $X:=\log$ $m(F)$ with

[TABLE]

We suppose that $X$ is a.s. finite.

We call a BPIRE $\mathbf{Y}$ supercritical if $\mathbf{E}[X]\in(0,+\infty],$ subcritical if $\mathbf{E}[X]\in[-\infty,0),$ and critical if either $\mathbf{E}[X]=0$ or $\mathbf{E}[X]$ does not exist.

It will be convenient to assume that if $Y_{n-1}=y_{n-1}>0$ is the population size of the ($n-1)$th generation of $\mathbf{Y}$ then first $\xi_{n1}+\ldots+\xi_{ny_{n-1}}$ individuals of the $n$th generation are born and than $\eta_{n}$ immigrants join the population.

This agreement allows us to consider a modified version $\mathbf{W}=\left\{W_{n},\ n\in\mathbb{N}_{0}\right\}$ of the process $\mathbf{Y}$ specified as follows. Assume, without loss of generality, that $Y_{0}>0.$ Let $W_{0}=Y_{0}$ and for $n\geq 1$,

[TABLE]

We call $\mathbf{W}$ a branching process with immigration stopped at zero and evolving in the random environment $\mathcal{E}$.

The aim of the present paper is to study, under the annealed approach, the tail distribution of the random variable

[TABLE]

for subcritical BPIRE. Observe that the tail distribution of $\zeta$ for critical BPIRE’s was described in [6].

With each pair of measures $(F,G)$ we associate the respective probability generating functions

[TABLE]

Given the environment $\mathcal{E}=\left\{(F_{n},G_{n}),n\in\mathbb{N}\right\}$, we construct the i.i.d. sequence of pairs of generating functions

[TABLE]

and use below the convolutions of the generating functions $F_{1},...,F_{n}$ specified for $0\leq i\leq n-1$ by the equalities

[TABLE]

We assume for convenience that $W_{0}=Y_{0}>0$ has the (random) probability generating function

[TABLE]

where $G_{0}(s)\overset{d}{=}G(s)$. Other cases of initial distributions may be considered in a similar way.

Denote

[TABLE]

and let

[TABLE]

It is known (see, Lemma 1 in [6]) that $\mathcal{R}(s)$ can be calculated by the formula

[TABLE]

The following restrictions are imposed on the distributions of $F$ and $G$.

Hypothesis A1. The BPRE is subcritical, i.e.

[TABLE]

and either $-\infty<\mathbf{E}\left[Xe^{X}\right]<0$ (the strongly subcritical case), or $\mathbf{E}\left[Xe^{X}\right]=0$ (the intermediate subcritical case), or there is a number $0<\beta<1$ such that

[TABLE]

(the weakly subcritical case).

Note that the subcritical BPRE’s mentioned in Hypothesis A1 do not exhaust all possible cases of subcritical BPRE’s. For instance, they do not include the subcritical BPRE’s where $\mathbf{E}\left[Xe^{tX}\right]=\infty$ for all $t>0$ (see [15]) or where $\mathbf{E}\left[Xe^{tX}\right]<0$ for all $0\leq t\leq\beta$ with $\beta=\sup\{t\geq 0:\mathbf{E}\left[e^{tX}\right]<\infty\}\in(0,1)$ (see [5]).

One of the main tools in analyzing properties of BPRE and BPIRE is a change of measure. We follow this approach and introduce a new measure $\mathbb{P}$ by setting, for any $n\in\mathbb{N}$ and any measurable bounded function $\psi:\Delta^{n}\times\mathbb{N}_{0}^{n+1}\rightarrow\mathbb{R}$

[TABLE]

with

[TABLE]

Here $\delta=1$ for strongly and intermediate subcritical BPIRE and $\delta=\beta$ for weakly subcritical BPIRE.

Observe that $\mathbf{E}[Xe^{\delta X}]=0$ translates into

[TABLE]

We assume that under the new measure the following set of conditions holds true.

Hypothesis A2. The distribution of $X$ is nonlattice. If a BPIRE is either intermediate or weakly subcritical then, with respect to $\mathbb{P}$, the distribution of $X$ belongs to the domain of attraction of a two-sided stable law with index $\alpha\in(1,2]$.

Since $\mathbb{E}[X]=0$ for the intermediate or weakly subcritical BPRE’s, Hypothesis A2 provides existence of an increasing sequence of positive numbers

[TABLE]

with slowly varying sequence $l_{1}(1),l_{1}(2),...$ such that the distribution law of $S_{n}/a_{n}$ converges weakly, as $n\rightarrow\infty$ to the mentioned two-sided stable law.

Our next assumption concerns the standardized truncated second moment of $F$,

[TABLE]

Define $\log^{+}x:=\log(\max(1,x))$.

Hypothesis A3.

1. If the BPRE is intermediate subcritical, then

[TABLE]

for some $\epsilon>0$ and some $a\in\mathbb{N}$.

2. If the BPRE is strongly subcritical, then

[TABLE]

for some $a\in\mathbb{N}$.

Now we impose restrictions on the immigration component.

**Hypothesis A4. **

[TABLE]

With Hypotheses A1-A4 in hands we are ready to formulate the main result of this note.

Theorem 1

Let Hypotheses A1-A4 be satisfied. Then, as $n\rightarrow\infty$

1) if the equation $r\mathcal{H}(r)=1$ has a root $1<r<\gamma^{-1}$, then

[TABLE]

2) if the BPIRE is weakly subcritical and $\gamma^{-1}\mathcal{H}(\gamma^{-1})<1,$ then

[TABLE]

with $b_{n}:=na_{n}$;

3) if the BPIRE is weakly subcritical and $\gamma^{-1}\mathcal{H}(\gamma^{-1})=1,$ then

[TABLE]

Remark 2

We show below that, under our conditions, the equation $r\mathcal{H}(r)=1$ always has a root $r\in(1,\gamma^{-1})$ for strongly and intermediate subcritical BPIRE.

We note that Zubkov [16] considered a similar problem for a Galton-Watson branching process with immigration $\{Y_{c}(n),n\geq 0\}$ evolving in the constant environment specified by probability generating functions $F(s)$ and $G(s),0\leq s\leq 1$. He investigated for this case the distribution of the so-called life period $\zeta_{c}$ initiated at time $N$ and defined as

[TABLE]

It was shown that if $G(0)>0$ and $F^{\prime}(1)<1$, i.e. the process $\{Y_{c}(n),n\geq 0\}$ is subcritical then $\mathbf{P}(\zeta>n)=(K+o(1))r_{c}^{n}$ as $n\to\infty$ where $r_{c}\in(0,1)$ is an explicitly known constant and the parameter $K$ is (depending on some additional technical conditions) either positive or equal to zero. Thus, the form (3) for the tail distribution of the random variable $\zeta$ in subcritical BPIRE’s is different from those known for the ordinary subcritical Galton-Watson processes with immigration.

The distribution of life periods for other models of branching processes with immigration evolving in a constant environment was analysed, for instance, in [4], [11], [13] and [14].

Theorem 1 complements the main result of [6] where it was shown that the tail distribution of $\zeta$ for a class of the critical BPIRE’s stopped at zero behaves like $n^{-\kappa}l(n).$ Here $\kappa\in(0,1)$ and $l(n)$ is a function slowly varying at infinity.

In the sequel if no otherwise is stated, we write $h_{n}\sim Ck_{n},C>0$ if $\lim_{n\rightarrow\infty}h_{n}/k_{n}=C$, $h_{n}=O(k_{n})$ if $\limsup_{n\rightarrow\infty}h_{n}/k_{n}<\infty,$ and $h_{n}=o(k_{n})$ if $\lim_{n\rightarrow\infty}h_{n}/k_{n}=0.$ We also denote by $C,C_{1},...$ positive constants which may vary from place to place.

2

Auxiliary results

Our goal is to investigate the asymptotic properties of $H_{n}$ and $H_{n}^{\ast}$ and, having the asymptotics in hands, to find an asymptotic representation for $R_{n}$ as $n\rightarrow\infty$. Observing that

[TABLE]

we reduce the first problem to considering the asymptotic behavior of $\overline{H}_{n}$. Similar reduction may be performed for $H_{n}^{\ast}$.

Set

[TABLE]

and denote

[TABLE]

Lemma 3

Let Hypotheses A1-A2 be satisfied. If the process is weakly subcritical, then for each $\varepsilon>0$, there exists $p=p(\varepsilon)$ such that

[TABLE]

for all sufficiently large $n$.

Proof Note that

[TABLE]

Therefore, for each $p\in\left[1,n/2\right]$

[TABLE]

We fix $k\in[p,n-p]$, set $S_{j}^{\ast}:=S_{k+j}-S_{k}$, $j=0,1,...,n-k$ and denote $L_{n-k}^{\ast}:=\min_{0\leq j\leq n-k}S_{j}^{\ast}$. The duality property of random walks gives

[TABLE]

According to Proposition 2.1 in [2], for each $\theta>0$ there exist positive constants $K_{i}=K_{i}(\theta),i=1,2$ such that, as $n\rightarrow\infty$

[TABLE]

We know by (2) that $b_{n}$ is a regularly varying sequence. Therefore, for any $\varepsilon>0$ there exists an integer number $p=p(\varepsilon)$ such that

[TABLE]

for all sufficiently large $n$.

The lemma is proved.

Lemma 4

Let Hypotheses A1-A2 be satisfied. If the process is intermediate subcritical, then for each $\varepsilon>0$, there exists $p=p(\varepsilon)$ such that

[TABLE]

for all sufficiently large $n,$ where $l_{2}(1),l_{2}(2),...$ is a sequence slowly varying at infinity.

Proof It follows from Lemma 2.2 in [3] that, as $n\rightarrow\infty$

[TABLE]

Setting $\beta=1$ in (4) and using the arguments of the preceding lemma we see that for any $\varepsilon>0$,

[TABLE]

completing the proof.

To go further we need to perform two more changes of measure using the right-continuous functions $U:\mathbb{R}$ $\rightarrow[0,\infty)$ and $V:\mathbb{R}$ $\rightarrow[0,\infty)$ specified by

[TABLE]

[TABLE]

It is known (see, for instance, [1] and [2]) that for any oscillating random walk

[TABLE]

[TABLE]

Let $\mathcal{E}=\left\{\mathbf{Q}_{1},\mathbf{Q}_{2},...\right\}$ be a random environment and let $\mathcal{F}_{n},n\geq 1,$ be the $\sigma$-field of events generated by the random vectors $\mathbf{Q}_{1},\mathbf{Q}_{2},...,\mathbf{Q}_{n}$ and the sequence $W_{0},W_{1},...,W_{n}$. The $\sigma$-fields $\{\mathcal{F}_{n},n\geq 1\}$ form a filtration $\mathfrak{F}$ and the increments $X_{n},n\geq 1$ of the random walk $\mathcal{S}$ are measurable with respect to the $\sigma$-field $\mathcal{F}_{n}$. We now introduce for each $n\geq 1$ a probability measure $\mathbb{P}_{(n)}^{+}$ on the $\sigma$-field $\mathcal{F}_{n}$ by means of the density

[TABLE]

In view of the martingale property (6) of $U$ the sequence of measures $\left\{\mathbb{P}_{(n)}^{+},n\geq 1\right\}$ is consistent on the filtration $\mathfrak{F}$. This and Kolmogorov’s extension theorem show that we may assume without loss of generality that there exists a probability measure $\mathbb{P}^{+}$ on $\mathfrak{F}$ such that

[TABLE]

In the sequel we allow for arbitrary initial value $S_{0}=x$. Then, we write $\mathbb{P}_{x}$ and $\mathbb{E}_{x}$ for the corresponding probability measures and expectations. Thus, $\mathbb{P}=\mathbb{P}_{0}.$ Using this agreement we rewrite (8) as

[TABLE]

for every $\mathcal{F}_{n}$-measurable random variable $O_{n}$.

Similarly, the martingale property (7) of $V$ gives rise to probability measures $\mathbb{P}^{-}_{x},x\leq 0$, and

[TABLE]

We now come back to branching processes. To have a unified approach in studying the asymptotic behavior of $H_{n}$ and $H_{n}^{\ast}$ as $n\rightarrow\infty$ we consider the sequence

[TABLE]

where $B(s)$ is a (random) probability generating function which is independent of the sequence $\mathbf{Q}_{n},n\geq 1$, and satisfies the restriction

Hypothesis A4.*

[TABLE]

Taking $B(s)=G_{0}(s)$ and $s=F_{n+1}(0)$ leads to $\gamma^{-n-1}H_{n},$ while $B(s)=(G_{0}(s)-G_{0}(0))/\left(1-G_{0}(0)\right)$ with the same $s$ gives $\gamma^{-n-1}H_{n}^{\ast}$.

Our plan is to find asymptotic representations of $B_{n}(s)$ for all types of subcritical BPIRE. To this aim we will use a decomposition

[TABLE]

where

[TABLE]

2.1 Weakly subcritical case

In this subsection we prove the following statement.

Theorem 5

Let Hypotheses A1-A2 and A4 be satisfied. If  the process is weakly subcritical with parameter $\beta\in(0,1),$ then for each $s\in(0,1)$*

[TABLE]

as $n\rightarrow\infty$.

The idea of proving Theorem 5 looks as follows. We show that, for a fixed $k$ and $n\rightarrow\infty$

[TABLE]

for some positive constants $C_{k}(s)$ and $\hat{C}_{k}(s)$, while $\sum_{k=p}^{n-p}B_{k,n}(s)$ is negligible in comparison with $1/b_{n}$ if $p$ is sufficiently large.

The proof of the asymptotic representations above is based on several important statements established in [2]. To check the applicability of the statements we need to prove several preparatory lemmas.

Let $Z(k,n)$ be the number of particles at moment $n$ in a branching process initiated at time $k$ by a single particle and $Z_{i}(k,n)$, $i=1,2,...$ be independent probabilistic copies of $Z(k,n)$.

Put

[TABLE]

where we assume (with a slight abuse of notation) that $B(s)$ is the probability generating function of $\eta_{0}$.

From now on we let $m:=\left[n/2\right],$ where $\left[x\right]$ stands for the integer part of $x$ and write

[TABLE]

Introduce two-dimensional random variables

[TABLE]

[TABLE]

and

[TABLE]

Lemma 6

If a BPIRE is weakly subcritical and Hypotheses A2 and A4 are valid then, for each $x\geq 0$*

[TABLE]

as $n\rightarrow\infty$, where $\left(\mathcal{U}_{\infty 1},\mathcal{U}_{\infty 2}\right)$ is a random vector whose components are positive with positive probabilities.

Proof Since the measure $\mathbb{P}_{x}^{+}$ imposes restriction on the offspring probability laws of particles but not on the reproduction of particles themselves, one can check that the random sequences

[TABLE]

form, correspondingly, a non-negative martingale and a submartingale with respect to the filtration $\mathfrak{F}$. Hence, there exists a random variable $\mathcal{U}_{\infty 1}$ such that, as $m\rightarrow\infty$

[TABLE]

Since

[TABLE]

the random variable $\mathcal{U}_{\infty 1}$ is positive with a positive probability.

Next, we claim that

[TABLE]

If we prove this statement, then we may conclude that, as $m\rightarrow\infty$

[TABLE]

where the random variable $\mathcal{U}_{\infty 2}$ is positive with a positive probability in view of $e^{-S_{m}}(\Xi(m)-Y(1,m))\geq e^{-S_{m}}Y(2,m).$

To establish the desired estimate recall that according to our change of measure,

[TABLE]

Conditioning first on the environment $\mathcal{E}$ and then on $\eta_{k-1},S_{0},S_{1},...S_{m}$ and observing that, for any $k<m$

[TABLE]

in view of (6), we obtain

[TABLE]

Since $U(y)$ is a renewal function, there exists a constant $C$ such that $U(y)\leq C(1+y)$ for all $y\geq 0$. Combining this estimate with the inequality

[TABLE]

we see that

[TABLE]

Recalling (5), it follows that, for all $m\in\mathbb{N}$

[TABLE]

as desired.

The lemma is proved.

Denote

[TABLE]

and introduce the random vector

[TABLE]

Setting $B(s)=s^{z}$ we obtain the following statement.

Corollary 7

Under the conditions of Lemma 6, for each $z\in\mathbb{N}$ and $x\geq 0$

[TABLE]

as $n\rightarrow\infty,$ where $\left(\mathcal{U}_{\infty 1}(z),\mathcal{U}_{\infty 2}\right)$ is a random vector whose components are positive with positive probabilities.

Now we deal with measure $\mathbb{P}^{-}$.

Lemma 8

If a BPIRE is weakly subcritical and Hypotheses A2 and A4 are valid then, for each fixed $s\in(0,1)$ and $x\leq 0$*

[TABLE]

as $n\rightarrow\infty$, where $\mathcal{V}_{\infty 1}(s)$ and $\mathcal{V}_{\infty 2}(s)$ are proper positive random variables.

Proof The fact that $\mathcal{V}_{n1}(s)\rightarrow\mathcal{V}_{\infty 1}(s)$ $\quad\mathbb{P}_{x}^{-}-a.s.$ as $n\rightarrow\infty$ is a particular case of Lemma 3.2 in [2]. To prove convergence of $\mathcal{V}_{n2}(s)$, note that given Hypotheses A2 and A4*,

[TABLE]

for every $s\in[0,1]$. Hence, for each $s>0$

[TABLE]

The lemma is proved.

For $u_{i}\geq 0,0\leq v_{i}\leq 1,t\geq 0\ (i=1,2)$ introduce the function

[TABLE]

One may check that $\varphi$ is bounded and continuous within the specified range of variables. For $z\in\mathbb{N}$, let

[TABLE]

where

[TABLE]

with scaling constant

[TABLE]

Lemma 9

If a BPIRE is weakly subcritical and Hypotheses A2 and A4 are valid then, for each $z\in\mathbb{N}$*

[TABLE]

Proof We write

[TABLE]

Observing that

[TABLE]

where $\widetilde{\mathcal{V}}_{n}(s)$ is the same as in (9) and using Theorem 2.7 in [2] we complete the proof of the lemma.

For $a>0$ and $u_{i}\geq 0,0\leq v_{i}\leq 1,z\leq 0\ (i=1,2)$ let

[TABLE]

Clearly, $\phi_{a}$ is bounded and continuous in the specified domain. By means of $\phi_{a}$ we specify, for $s\in[0,1]$ the function

[TABLE]

where

[TABLE]

with scaling constant

[TABLE]

Lemma 10

If a BPIRE is weakly subcritical and Hypotheses A2 and A4 are valid then, for each $s\in[0,1]$*

[TABLE]

Proof We write

[TABLE]

According (5)

[TABLE]

for all $n\in\mathbb{N}$ and $a>0$.

Further, we know from Lemmas 6–8 that the conclusion of Theorem 2.8 in [2] holds for $\phi_{a}(\mathbf{u},\mathbf{v},z)$, i.e.

[TABLE]

Hence, letting $a$ to infinity we prove the lemma.

Proof of Theorem 5 Since

[TABLE]

it follows from Lemma 3 that for any $\varepsilon>0$

[TABLE]

for all sufficiently large $n$ and $p=p(\varepsilon)$.

Further, for fixed $k\leq p$, we take the expectation with respect to the $\sigma-$algebra $\mathcal{F}_{k}$ and obtain

[TABLE]

where

[TABLE]

Using Lemma 9, applying the dominated convergence theorem and recalling (5), we conclude that

[TABLE]

Finally, we fix $j\geq 0$ and consider the expectation

[TABLE]

Denote $\overline{S}_{k}:=S_{n-j+k}-S_{n-j}\ (k=1,...,j)$ and let $\overline{\mathcal{F}}_{j}$ be the $\sigma$-algebra generated by $\mathbf{Q}_{n-j+1},...\mathbf{Q}_{n}$. Taking the internal expectation with respect to $\overline{\mathcal{F}}_{j}$ and supplying the respective variables with bars - we see that

[TABLE]

Using Lemma 10 and the dominated convergence theorem we conclude that

[TABLE]

Combining (10)-(11) with Proposition 2.1 in [2], we complete the proof.

2.2 Intermediate and strongly subcritical cases

In this subsection we find the asymptotics of $B_{n}(s)$ for intermediate and strongly subcritical BPIRE.

Theorem 11

Let Hypotheses A1-A3 and A4 be satisfied. If the process is intermediate subcritical then, as $n\rightarrow\infty$*

[TABLE]

Proof Recalling that

[TABLE]

we have, for fixed $j\geq 0$

[TABLE]

Using the duality property of random walks we see that

[TABLE]

where $\widehat{G}_{i}$ are independent probabilistic copies of $G_{i}$.

Then, for any $\kappa>0$,

[TABLE]

First observe that, as $n\rightarrow\infty$

[TABLE]

and, by monotonicity and Hypothesis A3

[TABLE]

where $\vartheta_{j},j=1,2,...$ are i.i.d. copies of $\vartheta$. Hence, there exists a positive random variable $\Theta(s)$ such that, as $n\to\infty$

[TABLE]

Using the arguments similar to those applied to prove Lemma 8, we conclude that, as $n\to\infty$

[TABLE]

Moreover,

[TABLE]

Hence it follows that

[TABLE]

for all $n$ and, according to Lemma 2.5 in [1], as $n\rightarrow\infty$

[TABLE]

By these estimates and the arguments similar to those applied to check the validity of Lemma 10, we conclude that, as $n\rightarrow\infty$

[TABLE]

Combining this result with Lemma 4 completes the proof.

Theorem 12

Let Hypotheses A1-A3 and A4 be satisfied. If the BPIRE is strongly subcritical then, for each $s\in(0,1)$*

[TABLE]

as $n\rightarrow\infty$.

Proof The proof is based on the transformed measure $\mathbb{P}$. In this case the inequality $\mathbf{E}[Xe^{X}]<0$ translates into

[TABLE]

Hence, the process is still subcritical under the probability measure $\mathbb{P}$ .

This fact, the equality

[TABLE]

the estimates

[TABLE]

and convergence

[TABLE]

as $n\rightarrow\infty,$ allow us to apply the dominated convergence theorem to conclude that

[TABLE]

The theorem is proved.

3 Proof of Theorem 1

Our proof of Theorem 1 essentially uses the following technical lemma.

Lemma 13

(see Theorem 1.4.6 in [17]) Let

[TABLE]

be a function with $\mathcal{T}_{n}\geq 0$ for all $n$. Assume that there exist a number $\varrho>1$ and a function $l_{0}(n)$ slowly varying at infinity such that

[TABLE]

as $n\rightarrow\infty$. If $\mathcal{C}(t)$ is an analytical function in a domain containing the circle

[TABLE]

then

[TABLE]

and

[TABLE]

Now everything is ready for proving Theorem 1. We know that

[TABLE]

Note that according to Theorems 5, 11, 12 and the change of measure (1) there is a positive constant $C\left(\delta\right)$ such that

[TABLE]

Besides, for $\gamma=\mathbf{E}\left[e^{\delta X}\right]$

[TABLE]

**Proof of Point 1) **Note that $\mathcal{H}(\gamma^{-1})=\infty$ for the strongly and intermediate subcritical cases. Hence a solution of the equation $r\mathcal{H}(r)=1$ within the interval $(1,\gamma^{-1})$ always exists for these cases and $\mathcal{H}^{\ast}(r)<\infty$. The same is true if $\gamma^{-1}\mathcal{H}(\gamma^{-1})>1$ for the weakly subcritical case. Taking these facts into account and recalling point 3) of Theorem 1 in ([7], XIII.10) we conclude that under the conditions of point 1) of Theorem 1, as $n\rightarrow\infty$

[TABLE]

**Proof of Point 2) ** Setting

[TABLE]

we see that, as $n\to\infty$

[TABLE]

If $\mathcal{T}(1)=\gamma^{-1}\mathcal{H}(\gamma^{-1})<1$ then, taking

[TABLE]

in Lemma 13 and writing

[TABLE]

we conclude that, as $n\to\infty$

[TABLE]

Observing that

[TABLE]

and using (12)–(13) we deduce, after evident estimates that

[TABLE]

as $n\rightarrow\infty$ as desired.

**Proof of Point 3) **Assume that $\mathcal{T}(1)=\gamma^{-1}\mathcal{H}(\gamma^{-1})=1$. Then

[TABLE]

By (12) and (13)

[TABLE]

Hence, applying to the recurrent sequence $\left\{\gamma^{-n}R_{n},n\geq 1\right\}$ point 2) of Theorem 1 in ([17], XIII.10), we conclude that

[TABLE]

Theorem 1 is proved.