From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment

Branching process in random environment (BPRE), initially introduced by Smith and Wilkinson [19], is a generalization of Galton-Watson process. The asymptotic behaviours for BPRE have been studied widely and intensively since the 1970s. For instance, one may refer to Athreya and Karlin [2, 1], Tanny [21, 22], Guivarc’h and Liu [8] and Birkner et al. [3] for some classical results on branching processes in independent and identically distributed (i.i.d.) environment or stationary and ergodic environment; Le Page and Ye [14] and Grama et al. [5, 6] for branching processes in Markovian environment. Regarding to the recent advances especially for supercritical branching processes in i.i.d. environment, for instance, Huang and Liu [11, 12] and Li et al. [15] established the large and moderate deviation principles, convergence theorems in $\mathbb{L}^{p}$ space and the central limit theorem (CLT); Grama et al. [7] studied asymptotic distribution and harmonic moments.

The main objective of this paper is to establish some additional limit theorems for supercritical BPRE, which have rarely appeared in the literature so far, including law of the iterated logarithm (LIL), strong law of large numbers (LLN), invariance principle and the rate of convergence in the CLT under Zolotarev and Wasserstein distances. The study of Wasserstein and Zolotarev distances are of independent interests, as they have been widely used in applications nowadays, such as machine learning ([9]), computer vision ([16], [20]) and microbiome studies ([23]). The methods used in this paper are mainly based on some recent development in martingale limit theory, especially on the invariance principles and LIL for martingales. One may refer to Hall and Heyde [10] and Shao [18] for the martingale limit theory and invariance principles; Wu et al. [24] for some recent advances on a Berry-Esseen bound for martingales.

Let’s firstly introduce the model of BPRE. Suppose $\xi=(\xi_{0},\,\xi_{1},\,\ldots)$ is a sequence of i.i.d. random variables. Usually, $\xi$ is called environment process and $\xi_{n}$ represents the random environment in the $n$-th generation. A discrete-time random process $(Z_{n})_{n\geqslant 0}$ is called BPRE $\xi$, if it satisfies the following recursive relation:

where $X_{n,i}$ represents the number of offspring produced by the $i$-th particle in the $n$-th generation. The distribution of $X_{n,i}$ depending on the environment $\xi_{n}$ is denoted by $p(\xi_{n})=\{p_{k}(\xi_{n})=\mathbb{P}(X_{n,i}=k|\xi_{n}):k\in\mathbb{N}\}.$ Suppose that given $\xi_{n}$, $(X_{n,i})_{i\geqslant 1}$ is a sequence of i.i.d. random variables; moreover, $(X_{n,i})_{i\geqslant 1}$ is independent of $(Z_{1},\ldots,Z_{n})$. Let $\left(\Gamma,\mathbb{P}_{\xi}\right)$ be the probability space under which the process is defined when the environment $\xi$ is given. The state space of the random environment $\xi$ is denoted by $\Theta$ and the total probability space can be regarded as the product space $\left(\Theta^{\mathbb{N}}\times\Gamma,\mathbb{P}\right),$ where $\mathbb{P}(\mbox{d}x,\mbox{d}\xi)=\mathbb{P}_{\xi}(\mbox{d}x)\tau(\mbox{d}\xi).$ That is, for any measurable positive function $g$ defined on $\Theta^{\mathbb{N}}\times\Gamma$, we have

where $\tau$ represents the distribution law of the random environment $\xi$. And $\mathbb{P}_{\xi}$ can be regarded as the conditional probability of $\mathbb{P}$ given the environment $\xi$. The expectations with respect to (w.r.t) $\mathbb{P}_{\xi}$ and $\mathbb{P}$ are denoted by $\mathbb{E}_{\xi}$ and $\mathbb{E}$, respectively. For any environment $\xi$, integer $n\geqslant 0$ and real number $p>0$, define

Then

Consider the following random variables

It is easy to see that

Then by recursion, we get $\Pi_{n}=\mathbb{E}_{\xi}Z_{n}$. Denote

The branching process $(Z_{n})_{n\geqslant 0}$ is called supercritical, critical or subcritical according to $\mu>0$, $\mu=0$ or $\mu<0$, respectively. Over all the paper, assume that

The condition (1.1) means that each particle has at least one offspring, which implies $X\geqslant 0$ a.s. and hence $Z_{n}\geqslant 1$ a.s. for any $n\geqslant 1$.

Let’s introduce now Zolotarev distance and Wasserstein distance respectively between two probability laws. For any $p>0$, denote by $[p]$ the largest integer which is strictly less than $p$, i.e.

Set $l=[p]$. Then $p$ can be decomposed uniquely as $p=l+\delta$, with $\delta\in(0,1]$. In the sequel, we will use this notation for the unique decomposition of any $p>0$. For a given $p>0$, consider $\Lambda_{p}$, the class of $l$-times continuously differentiable real-valued functions, defined by

For $l$-times continuously differentiable function $f$, set

Then for any $f\in\Lambda_{p}$,

Consider a set of probability laws $\mathcal{L}(\mu,\nu)$ on $\mathbb{R}^{2}$ with marginals $\mu$ and $\nu$. Zolotarev distance of order $p$ between $\mu$ and $\nu$ is defined by

For any $p>0$, the Wasserstein distance of order $p$ between $\mu$ and $\nu$ is defined by

Let $\mathcal{F}_{p}$ be the collection of all Borel probability measures on $\mathbb{R}$ with finite absolute moments of order $p$; then $(\mathcal{F}_{p},W_{p})$ forms a metric space, which is closely related to the topology of weak convergence of probability distributions on $\mathbb{R}$.

According to Kantorovich and Rubinstein [13], for any $0<p\leqslant 1$,

From Rio [17], it is proved that for any $p>1$,

where $c_{p}>0$ is a constant depending only on $p$.

Throughout the paper, $C$ denotes a positive constant whose value may vary from line to line. If $X$ and $Y$ are two random variables, such that they follow the probability laws $P_{X}$ and $P_{Y}$ respectively, then $\zeta_{p}(P_{X},\,P_{Y})$ (resp. $W_{p}(P_{X},\,P_{Y})$) is simply denoted by $\zeta_{p}(X,\,Y)$ (resp. $W_{p}(X,\,Y)$). We will use $G_{t}$ to represent the $\mathcal{N}(0,t)$-distributed random variable with variance $t>0$; in particular, $\mathcal{N}$ represents the standard normal random variable. And $a\vee b=\max\{a,b\}$, for any $(a,b)\in\mathbb{R}^{2}$.

In the sequel, denote by $X_{i}=\log m_{i},\ i\geqslant 0.$ Evidently, $(X_{i})_{i\geqslant 0}$ is a sequence of i.i.d. random variables depending only on the environment $\xi$. Let $(S_{n})_{n\geqslant 0}$ be the random walk associated with the branching process, which is defined as follows:

Then we have the following decomposition of $\log Z_{n}$:

where $W_{n}=\frac{Z_{n}}{\Pi_{n}}.$ The normalized population size $(W_{n})_{n\geqslant 0}$ is a non-negative martingale under both $\mathbb{P}$ and $\mathbb{P}_{\xi}$, w.r.t the natural filtration $(\mathcal{F}_{n})_{n\geqslant 0}$, defined by

By Doob’s martingale convergence theorem and Fatou’s lemma, we can obtain that $W_{n}$ converges a.s. to a finite limit $W$ and $\mathbb{E}W\leqslant 1$. We assume the following conditions throughout this paper:

The first condition above together with (1.1) imply in particular that

The second condition in (2.6) implies that $W_{n}$ converges to $W$ in $\mathbb{L}^{1}$ and

(See e.g. Tanny [22]). Therefore, it follows with the assumption (1.1) that $W>0$ and $Z_{n}\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}\infty$ a.s.

In the sequel, we will need the following conditions.

We have the following LIL for the process $(\log Z_{n})_{n\geqslant 0}$.

From the LIL theorem above, we can find immediately the following strong LLN for $(\log Z_{n})_{n\geqslant 0}$.

We also have the following invariance principle for the process $(\log Z_{n}-n\mu)_{n\geqslant 0}$, which only requires the existence of the second order moment $\mathbb{E}X^{2}<\infty$. And this has been insured in the assumption (2.6).

We have the following convergence rates of $(\log Z_{n})_{n\geqslant 0}$ in the CLT under Zolotarev and Wasserstein distances respectively.

The result above is an immediate consequence of Theorem 2.3, the equality (1.3) and the inequality (1.4) respectively for the cases $r\in[\delta,\,1]$ and $r\in(1,\,2]$.

Consider a sequence of martingale differences $(\varepsilon_{i})_{i\geqslant 1}$, defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, w.r.t a filtration $\mathcal{G}=(\mathcal{G}_{i})_{i\geqslant 0}$. Set

Then $(M_{n},\mathcal{G}_{n})_{n\geqslant 0}$ is a martingale. Set $V_{n}^{2}=\sum_{i=1}^{n}\varepsilon_{i}^{2}$ and $s_{n}^{2}=\mathbb{E}M_{n}^{2}$. We have the following lemma (see also Corollary 4.2 and Theorem 4.8 in Hall and Heyde [10]), from which Theorem 2.1 can be obtained.

Recall now that the random walk associated with the branching process

is the sum of i.i.d. random variables $(X_{i})_{i\geqslant 0}$. In the sequel, we will use the following notations. Let $\varepsilon_{i}=X_{i-1}-\mu$, $i\geqslant 1$ and

Then from (2.5), we have

Since $(\varepsilon_{i})_{i\geqslant 1}$ is a sequence of i.i.d. zero-mean random variables and $\sigma<\infty$, $(M_{n})_{n\geqslant 0}$ becomes a zero-mean and square integrable martingale w.r.t the filtration $(\mathcal{F}_{n})_{n\geqslant 0}$.

Theorem 2.2 can be proved by using Theorem 2.1 in Shao [18], where the following invariance principle for the martingale $(M_{n},\,\mathcal{G}_{n})_{n\geqslant 0}$ with the martingale difference sequence $(\varepsilon_{i})_{i\geqslant 1}$ is established.

To prove Theorem 2.3, the following lemmas will be used.