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Criticality for branching processes in random environment. (English) Zbl 1075.60107

Let $Z$ be a critical, discrete-time, one-type Markov branching process in a random environment ${\Pi }=\left({Q}_{1},{Q}_{2},{Q}_{3},\cdots \right)$, ${Q}_{k}$ i.i.d copies of $Q$, $Q$ having probability measures on $\left\{0,1,2,\cdots \right\}$ as its values. Define $m\left(Q\right):={\sum }_{y>0}yQ\left(\left\{y\right\}\right)$, $\zeta \left(a\right):={\sum }_{y\ge a}{y}^{2}Q\left(\left\{y\right\}\right)/m{\left(Q\right)}^{2}$, $a=0,1,2,\cdots$, let $S=\left({S}_{0},{S}_{1},{S}_{2},\cdots \right)$ be the random walk given by ${X}_{n}={S}_{n}-{S}_{n-1}:=logm\left({Q}_{n}\right)$. Suppose ${X}_{n}$ is a.s. finite and define $v\left(x\right):=1+{\sum }_{i>1}𝐏\left({S}_{\gamma \left(i\right)}\ge -x\right)$ when $x\ge 0$, and $v\left(x\right):=0$ otherwise, where the $\gamma \left(i\right)$ are the strict descending ladder epochs of $S$. Set ${X}_{t}^{r,n}:={Z}_{r+\left[\left(n-r\right)t\right]}/{\mu }_{r+\left[\left(n-r\right)t\right]}$, $0\le t<1$, where ${\mu }_{n}:=𝐄\left({Z}_{n},{Z}_{0},{\Pi }\right)$. Consider two sets of assumptions:

(A) There exits a $\rho$, $0<\rho <1$, such that $\left(1/n\right){\sum }_{1\le m\le n}𝐏\left({S}_{m}>0\right)<\rho$, $n\to \infty$ and, for some $\epsilon >0$ and some integer $a>0$, $𝐄\left({log}^{+}{\left(\zeta \left(a\right)\right)}^{1/\rho +\epsilon }<\infty$ and $𝐄\left[v\left({X}_{1}\right)\left({log}^{+}\zeta {\left(a\right)}^{1+\epsilon }\right]<\infty$.

(B) The distribution of ${X}_{1}$ belongs without centering to the domain of attraction of some stable law which is not one-sided and has index $\alpha$, $0<\alpha \le 2$, and, for some $\epsilon >0$ and some integer $a>0$, $𝐄{\left({log}^{+}\zeta \left(a\right)\right)}^{\alpha +\epsilon }<\infty$.

First, assume (A) or (B). Then, for some finite positive real number $\theta$,

$𝐏\left({Z}_{n}>0\right)\sim \theta 𝐏\left(min\left({S}_{1},{S}_{2},\cdots ,{S}_{n}\right)\ge 0\right),\phantom{\rule{1.em}{0ex}}n\to \infty ,$

and, as a corollary,

$𝐏\left({Z}_{n}>0\right)\sim \theta {n}^{-\left(1-\rho \right)}l\left(n\right),\phantom{\rule{1.em}{0ex}}n\to \infty ,$

with $l\left(n\right)$ slowly varying at infinity. For any sequence of integers $r\left(1\right),r\left(2\right),r\left(3\right),\cdots$ such that $r\left(n\right) and $r\left(n\right)\to \infty$, as $n\to \infty$, the conditional distribution of $\left({X}^{r\left(n\right),n}\mid {Z}_{n}>0\right)$ converges weakly (Skorokhod topology) to the law of a process with a.s. constant paths, $0<{W}_{t}<\infty$ a.s.. The distribution of $\left(min\left\{i\le n:{S}_{i}=min\left({S}_{0},\cdots ,{S}_{n}\right)\right\},min\left({S}_{0},\cdots ,{S}_{n}\right)\mid {Z}_{n}>0\right)$ converges weakly to a probability measure. Assuming (B), there exists a sequence $l\left(1\right),l\left(2\right),\cdots$ slowly varying at infinity, such that the conditional distribution of $\left({\left({n}^{-1/\alpha }l\left(n\right){S}_{\left[nt\right]}\right)}_{0\le t\le 1}\mid {Z}_{n}>0\right)$ converges weakly to the law of the meander of a strictly stable process with index, and, as a corollary, the corresponding result for $log{Z}_{\left[nt\right]}$ in place of ${S}_{\left[nt\right]}$ holds.

##### MSC:
 60J80 Branching processes 60G50 Sums of independent random variables; random walks 60F17 Functional limit theorems; invariance principles 60K37 Processes in random environments