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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 Π=(Q 1 ,Q 2 ,Q 3 ,), Q k i.i.d copies of Q, Q having probability measures on {0,1,2,} as its values. Define m(Q):= y>0 yQ({y}), ζ(a):= ya y 2 Q({y})/m(Q) 2 , a=0,1,2,, let S=(S 0 ,S 1 ,S 2 ,) be the random walk given by X n =S n -S n-1 :=logm(Q n ). Suppose X n is a.s. finite and define v(x):=1+ i>1 𝐏(S γ(i) -x) when x0, and v(x):=0 otherwise, where the γ(i) are the strict descending ladder epochs of S. Set X t r,n :=Z r+[(n-r)t] /μ r+[(n-r)t] , 0t<1, where μ n :=𝐄(Z n ,Z 0 ,Π). Consider two sets of assumptions:

(A) There exits a ρ, 0<ρ<1, such that (1/n) 1mn 𝐏(S m >0)<ρ, n and, for some ε>0 and some integer a>0, 𝐄(log + (ζ(a)) 1/ρ+ε < and 𝐄[v(X 1 )(log + ζ(a) 1+ε ]<.

(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 α, 0<α2, and, for some ε>0 and some integer a>0, 𝐄(log + ζ(a)) α+ε <.

First, assume (A) or (B). Then, for some finite positive real number θ,

𝐏(Z n >0)θ𝐏(min(S 1 ,S 2 ,,S n )0),n,

and, as a corollary,

𝐏(Z n >0)θn -(1-ρ) l(n),n,

with l(n) slowly varying at infinity. For any sequence of integers r(1),r(2),r(3), such that r(n)<n and r(n), as n, the conditional distribution of (X r(n),n Z n >0) converges weakly (Skorokhod topology) to the law of a process with a.s. constant paths, 0<W t < a.s.. The distribution of (min{in:S i =min(S 0 ,,S n )},min(S 0 ,,S n )Z n >0) converges weakly to a probability measure. Assuming (B), there exists a sequence l(1),l(2), slowly varying at infinity, such that the conditional distribution of ((n -1/α l(n)S [nt] ) 0t1 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 logZ [nt] in place of S [nt] holds.


MSC:
60J80Branching processes
60G50Sums of independent random variables; random walks
60F17Functional limit theorems; invariance principles
60K37Processes in random environments