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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
Citations:
Zbl 1075.60107
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