Limit theorems for a moderately subcritical branching process in a random environment

Abstract

Let {ξ,,} be a moderately subcritical branching process in a random environment with linear-fractional generating functions, m" be the conditional expectation of ξ,, with respect to the random environment. We prove theorems on convergence of the sequence of random processes {δ,,,ΐ/m,,,,,, / e (0,1)| ξ,, >0}as η -> °° in distribution, and of the initial and final segments of the random sequence ξο/mo, ξι/mi,..., ξ,,/m,, considered under the condition that {ξ,, > 0}.

A functional limit theorem for the logarithm of a moderately subcritical branching process in a random environment

“…In view of the constraints imposed on the random environment {π η }, the pairs of random variables (Χι,ηι), (^»ite), ... are independent and identically distributed. As was proved in [1], under the hypotheses of Theorem 1 the random walk satisfies the following conditions:…”

“…be identically distributed and independent for different n. We introduce the generating functions Λω=π1 0) + π1 1) ί +π1 2 ν + ... ) η€Ν 0 .…”

“…A sequence of non-negative integer-valued random variables {ξ η ,η € NO} is called a branching process in the random environment {π η , € NO} if In [1], where such a process was studied, it was shown that, if the generating functions f n (s), n € NO, are linear-fractional, Eexp \X\ |, E(T|I expXi) are finite, and the random variable X\ 9 if concentrated on a lattice, possesses an atom at zero, then, as η -)><*>, {ξΜ/^Η,ί€(0,1)|ξ Λ >0}Α^ (1) where m^ is the conditional mathematical expectation of ξ* under random environment, μ is a stochastic process on (0,1) whose realizations are positive constants, the symbol -> stands for the convergence in distribution in the space Z)[w, 1 -u] with Skorokhod topology [2] for any fixed u € (0,1/2). In this paper, basing on relation (1), we prove the following functional limit theorem.…”

“…In [1], it was established that under conditions (AMD) for any fixed j € N, £,/ <E N 0 , as n -> «>, {So>Si,... Λ;5 η _/,5 π _/+ι,... ,S n _i,S n |τ/-ι <n <τ/} W ^ ) c ) v /-l»** ' » °1 >°0 /» and _v/e c c .…”

Branching Processes in Random Environment

“…A list of older results is [Tan77], [Tan78], [CT84], [Gui85], [Tan88], [GZ91], [Ham92], [Afa93], [Koz95], [Liu96], [BV97], [DH97a], [Afa97], [VD97], [Afa98], [Afa01a], [VD02], [DGV04], [VD04].…”