A critical branching process with immigration in random environment¹¹1This work is supported by the Russian Science Foundation under grant 19-11-00111.

1. Introduction and statement of main result

Let $\left(\Omega,\mathcal{F},\mathbf{P}\right)$ be a probability space and $\Delta$ be the space of probability measures on $\mathbf{N}_{0}:=\left\{0,1,\ldots\right\}$ equipped with the metric of total variation. A random environment is a sequence of random elements $Q_{1},Q_{2},\ldots$, mapping the space $\left(\Omega,\mathcal{F},\mathbf{P}\right)$ into $\Delta^{2}$. Thus, $Q_{n}$ for each $n\in\mathbf{N}$ has the form $\left(F_{n},G_{n}\right)$, where $F_{n},G_{n}$ are probability measures on $\mathbf{N}_{0}$. A branching process with immigration in random environment ((BPIRE)) is a stochastic process possessing the following properties. For a fixed random environment $\left\{Q_{n},n\in\mathbf{N}\right\}$ this is an inhomogeneous branching Galton-Watson process with immigration (see [1], Chapter 6, § 7). Here, for each $n\in\mathbf{N}$, the number of immigrants joining the $\left(n-1\right)$th generation has the distribution $G_{n}$ and the offspring reproduction law of particles of the $\left(n-1\right)$th generation is $F_{n}$.

Let $Z_{n}$ be the size of $n$th generation without the immigrants which joined this generation (we assume that $Z_{0}=0$), $\eta_{n}$ be the number of immigrants which joined the $n$th generation. Let $f_{n}\left(\cdot\right)\$and $g_{n}\left(\cdot\right)$ be generating functions of distributions $F_{n}\$and $G_{n}$ respectively.

We consider this model under the assumption that the random elements $Q_{1},Q_{2},\ldots$ are independent and identically distributed. A more detailed definition of the BPIRE can be found in [2].

Set for $i\in\mathbf{N}$

(suppose that $0<f_{1}^{\prime}\left(1\right)<+\infty$, $0<g_{i}^{\prime}\left(1\right)<+\infty$ a.s.). Introduce the so-called associated random walk:

It is clear that the random vectors $\left(X_{1},\mu_{1}\right),\left(X_{2},\mu_{2}\right),\ldots$ are independent and identically distributed under our assumptions.

We impose the following restriction on the distribution of $X_{1}$.

Hypothesis A. The distribution of $X_{1}$ belongs without centering to the domain of attraction of some stable law with index $\alpha\in\left(0,2\right]$ and the limit law is not a one-sided stable law.

Under Hypothesis A the Skorokhod functional limit theorem is valid (see, for instance, [3], Chapter 16): there are such positive normalizing constants $C_{n}$ that, as $n\rightarrow\infty$,

where $W_{n}=\left\{C_{n}^{-1}S_{\left\lfloor nt\right\rfloor},\,t\geq 0\right\}$, the process $W=\left\{W\left(t\right),\,t\geq 0\right\}$ is a strictly stable Levy process with index $\alpha\in\left(0,2\right]$ and the symbol $\stackrel{{\scriptstyle D}}{{\to}}$ means convergence in distribution in the space $D\left[0,+\infty\right)$ with Skorokhod topology. Moreover,

where $\left\{l\left(n\right),n\in\mathbf{N}\right\}$ is a slowly varying sequence. It is known that the finite-dimensional distributions of the process $W$ are absolutely continuous. Note that $\rho:=\mathbf{P}\left(W\left(1\right)>0\right)\in\left(0,1\right)$ given Hypothesis A. Thus, the Spitzer-Doney condition is satisfied:

The Spitzer-Doney condition means that the random walk $\left\{S_{n}\right\}$ is oscillating. As result, the absolute values of its strict descending ladder heights constitute a renewal process with the corresponding renewal function $v\left(x\right)$, $x\geq 0$ (see [4] for a detailed definition of the function $v\left(\cdot\right)$). Similarly, weak ascending ladder heights of the random walk $\left\{S_{n}\right\}$ generate a renewal process with the corresponding renewal function $u\left(x\right)$, $x\geq 0$.

The aim of this paper is to prove a functional limit theorem for the process $\left\{Z_{\left\lfloor nt\right\rfloor},\,t\geq 0\right\}$, as $n\rightarrow\infty$ (see Theorem 1).

We need some notation and definitions to formulate the theorem. Let for $n\in\mathbf{N}$

It is known (see, for instance, [4], Lemma 2.5) that, if the Spitzer-Doney condition (2) is satisfied, then, as $n\rightarrow\infty$,

where $\left\{\left(Q_{i}^{+},S_{i}^{+},\mu_{i}^{+}\right)\right\}$, $\left\{\left(Q_{i}^{-},S_{i}^{-},\mu_{i}^{-}\right)\right\}$ are some random sequences. Moreover: a) the sequences $\left\{Q_{i}^{+},\,i\in\mathbf{N}\right\}$, $\left\{Q_{i}^{-},\,i\in\mathbf{N}\right\}$ can be viewed as some random environments; b) the sequences $\left\{S_{i}^{+},\,i\in\mathbf{N}\right\}$, $\left\{S_{i}^{-},\,i\in\mathbf{N}\right\}$ are the corresponding associated random walks ($S_{0}^{+}=S_{0}^{-}=0$); c) the sequences $\left\{\mu_{i}^{+},\,i\in\mathbf{N}\right\}$ and $\left\{\mu_{i}^{-},\,i\in\mathbf{N}\right\}$ are positive and constructed by $\left\{Q_{i}^{+},\,i\in\mathbf{N}\right\}$ and $\left\{Q_{i}^{-},\,i\in\mathbf{N}\right\}$, respectively, the same as the sequence $\left\{\mu_{i},\,i\in\mathbf{N}\right\}$ is constructed by $\left\{Q_{i},\,i\in\mathbf{N}\right\}$. Suppose that the sequences $\left\{Q_{i}^{+},\,i\in\mathbf{N}\right\}$, $\left\{Q_{i}^{-},\,i\in\mathbf{N}\right\}$ are defined on the same probability space $\left(\Omega^{\ast},\mathcal{F}^{\ast},\mathbf{P}^{\ast}\right)$ and are independent (below we denote the expectation on this probability space by $\mathbf{E}^{\ast}$).

We now come back to our initial BPIRE. Set $\mathbf{N}_{i}=\left\{i,i+1,\ldots\right\}$ for $i\in\mathbf{Z}$. Fix $i\in\mathbf{N}_{0}$ and, for $n\in\mathbf{N}_{i}$, denote by $Z_{i,n}$ the total number of particles in the $n$th generation which are the descendants of the immigrants joined the $i$th generation (we assume that $Z_{i,n}=0$ for $i\geq n$ and $i<0$). Note that the random sequence $\left\{\eta_{i};\,Z_{i,n},\,n\in\mathbf{N}_{i+1}\right\}$ is a usual (without immigration) branching process in the random environment $\left\{G_{i+1};\,F_{n},\,n\in\mathbf{N}_{i+1}\right\}$. In particular, if the random environment is fixed, then $G_{i+1}$ is the distribution of the random variable $\eta_{i}$ which should be interpreted as the number of particles in the initial generation. Set for $n\in\mathbf{N}_{i}$

The sequence $\left\{\eta_{i};\,a_{i,n}Z_{i,n},\,n\in\mathbf{N}_{i+1}\right\}$ is a nonnegative martingale if the random environment $\left\{G_{i+1};\,F_{n},\,n\in\mathbf{N}_{i+1}\right\}$ is fixed. Hence (without assuming that the random environment is fixed), there is a finite limit $\lim_{n\rightarrow\infty}a_{i,n}Z_{i,n}$ $\mathbf{P}$-a.s.

Set

The sequence $\mathcal{E}^{\ast}:=\left\{Q_{k}^{\ast},\,k\in\mathbf{Z}\right\}$ can be considered as a random environment (we denote the components of $Q_{k}^{\ast}$ by $G_{k}^{\ast}\$and $F_{k}^{\ast}$). We assume that the probability space $\left(\Omega^{\ast},\mathcal{F}^{\ast},\mathbf{P}^{\ast}\right)$ is reach enough for we are able to define on it a branching process with immigration in the random environment $\mathcal{E}^{\ast}$. Fix $i\in\mathbf{Z}$ and, for $j\in\mathbf{N}_{i}$, denote by $Z_{i,j}^{\ast}$ the total number of particles in the $j$th generation being descendants of immigrants which joined the $i$th generation (we denote the number of such immigrants as $\eta_{i}^{\ast}$). Note that the sequence $\left\{\eta_{i}^{\ast};\,Z_{i,j}^{\ast},\,j\in\mathbf{N}_{i+1}\right\}$ is a branching process in the random environment $\left\{G_{i+1}^{\ast};\,F_{j}^{\ast},\,j\in\mathbf{N}_{i+1}\right\}$ with the initial value $\eta_{i}^{\ast}$. The sequence $\left\{S_{j}^{\ast}-S_{i}^{\ast},\,j\in\mathbf{N}_{i}\right\}$ is the associated random walk and the random variable $\mu_{i}^{\ast}$ is under fixed environment the mean of the random variable $\eta_{i}^{\ast}$. Set

In accordance with the above the limit

exists $\mathbf{P}^{\ast}$-a.s. and $\mathbf{P}^{\ast}\left(\zeta_{i}^{\ast}>0\right)>0$ for $i\in\mathbf{N}_{0}$ (see [4], Proposition 3.1).

Introduce the following random series:

It is clear that $\Sigma_{1}>0$ $\mathbf{P}^{\ast}$-a.s. and $\mathbf{P}^{\ast}\left(\Sigma_{2}>0\right)>0$. Both series converge $\mathbf{P}^{\ast}$-a.s. under certain restrictions (see Lemma 4).

Let $W$ be a strictly stable Levy process with index $\alpha$ (in the sequel we call $W$ simply the Levy process). By the Levy process we specify the (lower) level $L=\left\{L\left(t\right),\,t\geq 0\right\}$ of the Levy process as

Let, further, $\gamma_{1},\gamma_{2},\ldots$ be an independent of $W$ sequence of independent random variables distributed as the random variable $\Sigma_{2}/\Sigma_{1}$.

By these ingredients we define finite-dimensional distributions of a random process $Y=\left\{Y(t),\,t\geq 0\right\}$ which plays an important role in the sequel. First we set $Y(0)=0$. Consider an arbitrary $m\in\mathbf{N}$ and arbitrary moments $t_{1},t_{2},\ldots,t_{m}$: $0=t_{0}<t_{1}<t_{2}<\ldots<t_{m}$. The random vector $\left\{Y(t_{1}),\ldots,Y(t_{m})\right\}$ coincides in distribution with the following vector $\widehat{Y}:=\left\{\widehat{Y}_{1},\ldots,\widehat{Y}_{m}\right\}$. We describe at first the possible values of the vector $\widehat{Y}$. Its first several coordinates coincide with $\gamma_{1}$, the next several coordinates coincide with $\gamma_{2}$ and so on up to the $m$th coordinate. The coordinates of the vector $\widehat{Y}$ are specified according to the level $L$ of the Levy process $W$. The first coordinate $\widehat{Y}_{1}$ is equal to $\gamma_{1}$. Let the coordinate $\widehat{Y}_{k}$ for some $k<m$ be known. For instance, $\widehat{Y}_{k}=\gamma_{l}$ for some $l\in\mathbf{N}$. If the level of the Levy process at the moment $t_{k+1}$ remains the same as at moment $t_{k}$, i.e. $L\left(t_{k+1}\right)=L\left(t_{k}\right)$, then $\widehat{Y}_{k+1}=\gamma_{l}$. If the level of the Levy process at the moment $t_{k+1}$ is changed, i.e. $L\left(t_{k+1}\right)<L\left(t_{k}\right)$, then $\widehat{Y}_{k+1}=\gamma_{l+1}$.

Set for $n\in\mathbf{N}_{0}$

Introduce for each $n\in\mathbf{N}$ the random process $Y_{n}=\left\{Y_{n}\left(t\right),\,t\geq 0\right\}$, where

Note that for $k\in\mathbf{N}$ the ratio $b_{k}/a_{k}$ is equal to the mean of $Z_{k}$ for a fixed random environment.

Let the symbol $\Rightarrow$ means convergence of random processes in the sense of finite-dimensional distributions and $\ln^{+}x=\max\left(0,\ln x\right)$ for $x>0$.

Theorem 1. If Hypothesis A is valid and $\mathbf{E}\left(\ln^{+}\mu_{1}\right)^{\alpha+\varepsilon}<+\infty$ for some $\varepsilon>0$, then, as $n\rightarrow\infty$,

A detailed description of the theory of critical (when Hypothesis A is valid) branching processes in random environment is available in [4] and [5].

A particular case of a subcritical BPIRE (when the offspring generating function $f_{n}\left(\cdot\right)$ is fractional-linear and $g_{n}\left(s\right)\equiv s$ for each $n\in\mathbf{N}$) was considered in [6]. The main attention there was paid to obtaining an exponential estimate for the tail distribution of the so-called life period of this process (i.e., the time until the first extinction). A more general class of subcritical BPIRE was analyzed in [7] where a limit theorem describing the population size at a distant moment was proved and an exponential estimate for the tail distribution of the life period was established. A strong law of large numbers and a central limit theorem for a wide class of subcritical BPIRE were proved in [8].

A critical BPIRE was considered in [9] where sufficient conditions of transience and recurrence were obtained. The author of [10], studying a random walk in random environment, proved a particular case of Theorem 1 (when the offspring generating function $f_{n}\left(\cdot\right)$ is fractional-linear and $g_{n}\left(s\right)\equiv s$ for each $n\in\mathbf{N}$). We would like to stress that the proof used in the present paper differs significantly from that one given in [10]. We also mention the papers [11], [12] and [13] in which critical and supercritical processes (with stopped immigration) are considered under some restrictions on their lifetime.

Recent papers [2] and [14] contain exact asymptotic formulae for the tail distribution of the life period for critical and subcritical BPIRE.

2. Auxiliary statements

Let $\tau_{n}$ be the first moment when the minimum of the random walk $S_{0},\ldots,S_{n}$ is attained:

Set for $n\in\mathbf{N}$

For positive integers numbers $n_{1}<n_{2}$ set

Lemma 1. If the Spitzer-Doney condition (2) is satisfied, then, as $n\rightarrow\infty$,

Proof. We demonstrate for simplicity only convergence of one-dimensional distributions. Fix $i\in\mathbf{N}_{0}$. Let $A$ be a one-dimensional $S_{i}^{\ast}$-continuous (relative to the measure $\mathbf{P}^{\ast}$) Borel set. Then for $n\geq i$

and by the Markov property of random walks we have that

Thus,

If the Spitzer-Doney condition is satisfied, then the following generalized arcsine law is valid (see, for instance, [15], Chapter 8, Theorem 8.9.9): for $x\in\left[0,1\right]$

We pass to the limit in formula (7), as $n\rightarrow\infty$. Due to (8)

Therefore the limit of the left-hand side of (7) coincides with the limit of probability $\mathbf{P}\left(S_{i,n}^{\prime}\in A\right)$, as $n\rightarrow\infty$, if at least one of these limits exists.

If $\varepsilon\in\left(0,1\right)$ and $n$ is large enough, then by (7)

where

Clearly,

Therefore

In view of (3) the probability $\mathbf{P}\left(\left.S_{i}\in A\,\right|\,L_{n-k}\,\geq 0\right)$ tends, as $n\rightarrow\infty$, to $\mathbf{P}^{\ast}\left(S_{i}^{+}\in A\,\right)=\mathbf{P}^{\ast}\left(S_{i}^{\ast}\in A\,\right)$ uniformly over $0\leq k\leq\left\lfloor\left(1-\varepsilon\right)n\right\rfloor$. Consequently,

implying

It follows from relations (9)-(11) that for $i\in\mathbf{N}_{0}$

Thus,

We now fix $i\in\mathbf{N}$. Let $A$ be a one-dimensional $S_{-i}^{\ast}$-continuous (relative to the measure $\mathbf{P}^{\ast}$) Borel set. Then for $n\geq i$

and by the Markov property and the duality property of random walks we have that

Thus,

therefore, if $\varepsilon\in\left(0,1\right)$ and $n$ is large enough, then

where

It is not difficult to show (see our proof of relation (10)) that

Due to (4) the probability $\mathbf{P}\left(\left.-S_{-i}\in A\,\right|\,M_{k}\,<0\right)$ tends, as $n\rightarrow\infty$, to $\mathbf{P}^{\ast}\left(-S_{-i}^{-}\in A\,\right)=\mathbf{P}^{\ast}\left(S_{-i}^{\ast}\in A\,\right)$ uniformly over $\left\lfloor\varepsilon n\right\rfloor<k\leq n$. Therefore

As result, we obtain that

proving (12) for $i\in\mathbf{Z\backslash N}_{0}$. Thus, convergence of one-dimensional distributions in (6) is established.

The lemma is proved.

Remark 1. It is not difficult to verify (see [16], Lemma 1) that relation (6) admits the following generalization: for any $a\leq 0$ and $b>0$, as $n\rightarrow\infty$,

Recall that $\left(\Omega,\mathcal{F},\mathbf{P}\right)$ is the underlying probability space. Set

Let $\mathcal{F}_{n}$, $n\in\mathbf{N}$, denote the $\sigma$-algebra generated by the segment of the random environment $Q_{1},\ldots,Q_{n}$ and the random variables $Z_{i,j}$ for $\left(i,j\right)\in I_{n}^{\left(2\right)}$. We now introduce a probability measure $\mathbf{P}^{+}$ on the $\sigma$-algebra $\mathcal{F}_{\infty}:=\sigma\left(\cup_{n=1}^{\infty}\mathcal{F}_{n}\right)$, defined for each $n\in\mathbf{N}_{0}$ and each $\mathcal{F}_{n}$-measurable nonnegative random variable $\beta$ by the formula

This may require a change of the underlying probability space (see [4] for more details). Similarly, we also introduce a probability measure $\mathbf{P}^{-}$ on the $\sigma$-algebra $\mathcal{F}_{\infty}$, defined for each $n\in\mathbf{N}_{0}$ and each $\mathcal{F}_{n}$-measurable nonnegative random variable $\beta$ by the formula

Recall that the functions $v\left(\cdot\right)$ and $u\left(\cdot\right)$ in formulae (14) and (15) are defined after relation (2). Thus, three measures $\mathbf{P},\mathbf{P}^{+},\mathbf{P}^{-}$ are defined on one and the same measurable space $\left(\Omega,\mathcal{F}_{\infty}\right)$. To explicitly indicate the measure on $\left(\Omega,\mathcal{F}_{\infty}\right)$ according to which we consider this or those random elements we use the measure symbol as a lower index.

For instance, it is shown in Lemma 2.5 from [4] that

(the lower index $\mathbf{P}^{+}$ for a random sequence shows here that the measure $\mathbf{P}^{+}$ is used on the space $\left(\Omega,\mathcal{F}_{\infty}\right)$). Similarly,

Due to (16), (17) and our assumption about the independence of the left-hand sides of these relations, the product of probability spaces $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{+}\right)$ and $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{-}\right)$ may be considered as a probability space $\left(\Omega^{\ast},\mathcal{F}^{\ast},\mathbf{P}^{\ast}\right)$ and, consequently, the direct product of the measures $\mathbf{P}^{+}$ and $\mathbf{P}^{-}$ may be treated as the measure $\mathbf{P}^{\ast}$.

Remark 2. If a random element $\xi$ is given on the space $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{+}\right)$ we can define the random element $\xi^{+}$, specified on the product of the spaces $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{+}\right)$ and $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{-}\right)$ by means of the formula $\xi^{+}\left(\omega_{1},\omega_{2}\right)=\xi\left(\omega_{1}\right)$ for $\left(\omega_{1},\omega_{2}\right)\in\Omega\times\Omega$. It is clear that $\mathbf{P}^{\ast}\left(\xi^{+}\in A\right)=\mathbf{P}^{+}\left(\xi\in A\right)$ for an arbitrary one-dimensional Borel set $A$. Similarly, if a random element $\xi$ is given on the space $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{-}\right)$ we can define the random element $\xi^{-}$, specified on the product of the spaces $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{+}\right)$ and $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{-}\right)$ by means of the formula $\xi^{-}\left(\omega_{1},\omega_{2}\right)=\xi\left(\omega_{2}\right)$ for $\left(\omega_{1},\omega_{2}\right)\in\Omega\times\Omega$, and $\mathbf{P}^{\ast}\left(\xi^{-}\in A\right)=\mathbf{P}^{-}\left(\xi\in A\right)$ for an arbitrary one-dimensional Borel set $A$. In accordance with the agreement we can consider the random elements standing in the left-hand sides of formulae (16) and (17) as generated by the random elements $\left\{\left(Q_{i},S_{i},\mu_{i}\right),\,i\in\mathbf{N}\right\}_{\mathbf{P}^{+}}$ and $\left\{\left(Q_{i},S_{i},\mu_{i}\right),\,i\in\mathbf{N}\right\}_{\mathbf{P}^{-}}$ respectively.

Lemma 2. If the Spitzer-Doney condition (2) is satisfied, then, as $n\rightarrow\infty$,

where $\left\{\zeta_{i}^{\ast},\,i\in\mathbf{N}_{0}\right\}$ is the random sequence defined by relation (5).

Proof. By virtue of the first part of Lemma 2.5 from [4] for $k\in\mathbf{N}$, as $n\rightarrow\infty$,

Note (see [4], Section 3) that in view of (14) for a fixed $i\in\mathbf{N}_{0}$ the random sequence $\left\{\eta_{i};\,Z_{i,j},\,j\in\mathbf{N}_{i+1}\right\}_{\mathbf{P}^{+}}$ given on the probability space $\left(\Omega,\mathcal{F}_{\infty},\mathbf{P}^{+}\right)$ is a branching process in the random environment $\left\{G_{i+1};\,F_{n},\,n\in\mathbf{N}_{i+1}\right\}_{\mathbf{P}^{+}}$. Hence, if the random environment is fixed, the sequence $\left\{\eta_{i};\,a_{i,j}Z_{i,j},\,j\in\mathbf{N}_{i+1}\right\}_{\mathbf{P}^{+}}$ is a non-negative martingale. Because of this (without assuming that the random environment is fixed) there is $\mathbf{P}^{+}$-a.s. the finite limit

It means, in view of the second part of Lemma 2.5 from [4], that, as $n\rightarrow\infty$,

To prove relation (18), it remains to note that in view of Remark 2

The lemma is proved.

Set for $n\in\mathbf{N}$

Lemma 3. If the Spitzer-Doney condition (2) is satisfied, then, as $n\rightarrow\infty$,

where $\left\{\zeta_{i}^{\ast},\,i\in\mathbf{N}_{0}\right\}$ is the random sequence defined by relation (5).

Proof. We demonstrate for simplicity only convergence of one-dimensional distributions. Fix $i\in\mathbf{N}_{0}$. Let $A$ be an arbitrary one-dimensional $\zeta_{i}^{\ast}$-continuous (relative to the measure $\mathbf{P}^{\ast}$) Borel set. Then for $n\geq i$