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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 $\Pi= (Q_1, Q_2, Q_3,\dots)$, $Q_k$ i.i.d copies of $Q$, $Q$ having probability measures on $\{0,1,2,\dots\}$ as its values. Define $m(Q):= \sum_{y> 0}yQ(\{y\})$, $\zeta(a):= \sum_{y\ge a}y^2Q(\{y\})/m(Q)^2$, $a= 0,1,2,\dots$, let $S =(S_0, S_1,S_2,\dots)$ be the random walk given by $X_n= S_n- S_{n-1}:= \log m(Q_n)$. Suppose $X_n$ is a.s. finite and define $v(x):= 1+ \sum_{i> 1}{\bold P}(S_{\gamma(i)}\ge -x)$ when $x\ge 0$, and $v(x):= 0$ otherwise, where the $\gamma(i)$ are the strict descending ladder epochs of $S$. Set $X^{r,n}_t:= Z_{r+[(n- r)t]}/\mu_{r+[(n- r)t]}$, $0\le t< 1$, where $\mu_n:= {\bold E}(Z_n, Z_0,\Pi)$. Consider two sets of assumptions: (A) There exits a $\rho$, $0< \rho< 1$, such that $(1/n)\sum_{1\le m\le n}{\bold P}(S_m> 0)<\rho$, $n\to \infty$ and, for some $\varepsilon> 0$ and some integer $a> 0$, ${\bold E}(\log^+(\zeta(a))^{1/\rho+\varepsilon}<\infty$ and ${\bold E}[v(X_1)(\log^+\zeta(a)^{1+\varepsilon}]<\infty$. (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 $\varepsilon> 0$ and some integer $a> 0$, ${\bold E}(\log^+\zeta(a))^{\alpha+\varepsilon}< \infty$. First, assume (A) or (B). Then, for some finite positive real number $\theta$, $${\bold P}(Z_n> 0)\sim\theta{\bold P}(\min(S_1,S_2,\dots,S_n)\ge 0), \quad n\to\infty,$$ and, as a corollary, $${\bold P}(Z_n> 0)\sim\theta n^{-(1-\rho)}l(n), \quad n\to\infty,$$ with $l(n)$ slowly varying at infinity. For any sequence of integers $r(1),r(2),r(3),\dots$ such that $r(n)< n$ and $r(n)\to\infty$, as $n\to \infty$, the conditional distribution of $(X^{r(n),n}\mid Z_n> 0)$ converges weakly (Skorokhod topology) to the law of a process with a.s. constant paths, $0< W_t<\infty$ a.s.. The distribution of $(\min\{i\le n: S_i= \min(S_0,\dots, S_n)\},\min(S_0,\dots, S_n)\mid Z_n> 0)$ converges weakly to a probability measure. Assuming (B), there exists a sequence $l(1),l(2),\dots$ slowly varying at infinity, such that the conditional distribution of $((n^{-1/\alpha}l(n) S_{[nt]})_{0\le t\le 1}\mid Z_n> 0)$ converges weakly to the law of the meander of a strictly stable process with index, and, as a corollary, the corresponding result for $\log Z_{[nt]}$ in place of $S_{[nt]}$ holds.

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