Critical branching processes evolving in an unfavorable random environment

2022/09/27 by Vatutin, Vladimir, Dyakonova, Elena · 1 citation
#60F17 #60G50 #60J80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2209.13611

Abstract

Let \ Z_n,n=0,1,2,...\ be a critical branching process in random environment and let \ S_n,n=0,1,2,...\ be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a sequence a₁,a₂,..., slowly varying at infinity such that the conditional distributions P( \fracS_na_n≤ x|Z_ngt;0) , x∈ (-∞ ,+∞ ),% weakly converges, as n→ ∞ to the distribution of a strictly positive and proper random variable. In this paper we supplement this result with a description of the asymptotic behavior of the probability P( S_n≤ φ(n);Z_ngt;0) ,% if φ(n)→ ∞ as n→ ∞ in such a way that φ(n)=o(a_n).

Cited by

• Limit theorems for critical branching processes in an extremely unfavorable random environment

Related

• Some functionals for random walks and critical branching processes in extreme random environment
• Limit theorems for critical branching processes in an extremely unfavorable random environment
• Reduced critical branching processes in non-favorable random environment
• Random walks conditioned to stay non-negative and branching processes in non-favorable random environment
• Path to survival for the critical branching processes in a random\n environment