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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 \(m(Q):= \sum_{y> 0}yQ(\{y\})\), \(\zeta(a):= \sum_{y\geq a}y^2Q(\{y\})/m(Q)^2\), \(a= 0,1,2,\dots\), 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 \(v(x):= 1+ \sum_{i> 1}{\mathbf P}(S_{\gamma(i)}\geq -x)\) when \(x\geq 0\), and \(v(x):= 0\) otherwise, where the \(\gamma(i)\) are the strict descending ladder epochs of \(S\). Set \(X^{r,n}_t:= Z_{r+[(n- r)t]}/\mu_{r+[(n- r)t]}\), \(0\leq t< 1\), where \(\mu_n:= {\mathbf E}(Z_n, Z_0,\Pi)\). Consider two sets of assumptions:
(A) There exits a \(\rho\), \(0< \rho< 1\), such that \((1/n)\sum_{1\leq m\leq n}{\mathbf P}(S_m> 0)<\rho\), \(n\to \infty\) and, for some \(\varepsilon> 0\) and some integer \(a> 0\), \({\mathbf E}(\log^+(\zeta(a))^{1/\rho+\varepsilon}<\infty\) and \({\mathbf E}[v(X_1)(\log^+\zeta(a)^{1+\varepsilon}]<\infty\).
(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 \(\alpha\), \(0<\alpha\leq 2\), and, for some \(\varepsilon> 0\) and some integer \(a> 0\), \({\mathbf E}(\log^+\zeta(a))^{\alpha+\varepsilon}< \infty\).
First, assume (A) or (B). Then, for some finite positive real number \(\theta\), \[ {\mathbf P}(Z_n> 0)\sim\theta{\mathbf P}(\min(S_1,S_2,\dots,S_n)\geq 0), \quad n\to\infty, \] and, as a corollary, \[ {\mathbf P}(Z_n> 0)\sim\theta n^{-(1-\rho)}l(n), \quad n\to\infty, \] 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 \((\min\{i\leq n: S_i= \min(S_0,\dots, S_n)\},\min(S_0,\dots, S_n)\mid Z_n> 0)\) 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 \(((n^{-1/\alpha}l(n) S_{[nt]})_{0\leq t\leq 1}\mid Z_n> 0)\) 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 (Galton-Watson, birth-and-death, etc.)
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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