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Limit theorems for weakly subcritical branching processes in random environment. (English) Zbl 1262.60083
Let $$Z= (Z_n)_{n\geq 0}$$ be a discrete-time one-type branching process with random environment $$\Pi= \{Q_1,Q_2,\dots\}$$, $$Q_\nu$$ i.i.d. $$\sim Q$$, $$Z_0$$ independent of $$\Pi$$. Denote by $$\Delta$$ the set of probability measures on $$\{0,1,2,\dots\}$$. The asymptotic behaviour of $$Z$$ is determined mainly by the random walk $$S= (S_n)_{n\geq 0}$$, $$S_0= 0$$, $$S_n= S_{n-1}+ X_n$$, $$X_n:= \log m(Q_n)\sim X$$, $$n> 0$$, $$m(q):= \sum_{y> 0} yq(\{y\}),\;q\in\Delta$$. The process $$Z$$ is sub-critical if $$S_n$$ drifts to $$-\infty$$. It is called weakly sub-critical if there is a number $$\beta\in(0,1)$$ such that $$\operatorname{E}[Xe^{\beta X}]= 0$$. Suppose that $$Z$$ is weakly sub-critical and that the distribution of $$X$$ is non-lattice and has finite variance (or, more generally, is in the domain of attraction of a stable law with index $$\alpha\in(1,2]$$), and that, for some $$\varepsilon> 0$$ and some $$a\in\mathbb{N}$$, $\operatorname{E}\Biggl(\log\max\Biggl(1, \sum_{y\geq a} y^2Q(\{y\})/m(Q)^2\Biggr)\Biggr)^{a+\varepsilon}< \infty.$ Then there exist $$\kappa,\kappa'\in(0,\infty)$$ such that $\operatorname{P}(Z_n> 0)\sim\kappa\operatorname{P}[\min(S_1,\dotsc, S_n)> 0]$ and $\operatorname{P}(Z_n> 0)\sim\kappa'(\operatorname{E}[e^{\beta X}])^n/na_n,$ as $$n\to\infty$$, where $$a_n= n^{1/\alpha}c_n$$, with $$c_1,c_2,\dots$$ a slowly varying sequence such that $$\operatorname{P}(S_n/a_n\in dx)\to s(x)\,dx$$ weakly, as $$n\to\infty$$, where $$s(x)$$ is the density of the limiting stable law.
Furthermore, the conditional laws $$\mathfrak{L}(Z_n| Z_n> 0)$$, $$n\geq 1$$, converge weakly to a probability distribution on $$\mathbb{N}$$, and the sequence $$\operatorname{E}(Z_n^\theta| Z_n> 0)$$ is bounded for every $$\theta<\beta$$, implying the convergence to the corresponding moment of the limit distribution. Finally, there is a process $$\{W_t ; t\in[0,1]\}$$ such that, as $$t\to\infty$$, $\mathfrak{L}(\exp(- S_{r(n)+ [(n-2r(n))t]} Z_{r(n)+ [(n-r(n))t]}),\;t\in [0,1]|Z_n> 0)\Rightarrow\mathfrak{L}(W_t, t\in [0,1])$ weakly in the Skorokhod space $$D[0,1]$$, where $$(r(n))_{n>0}$$ is a sequence of natural numbers, $$r(n)\to\infty$$, and there is a random variable $$W$$ such that $$W_t= W$$ a.s. for all $$t\in[0,1]$$, $$\operatorname{P}(0< W<\infty)= 1$$. The paper is methodically related to the preceding paper on critical branching processes with random environment by V. I. Afanasyev et al. [Ann. Probab. 33, No. 2, 645–673 (2005; Zbl 1075.60107)].

##### MSC:
 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60K37 Processes in random environments 60G50 Sums of independent random variables; random walks 60F17 Functional limit theorems; invariance principles
Zbl 1075.60107
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