How many families survive for a long time

TL;DR

This paper analyzes the long-term survival probabilities of families in a critical branching process within a random environment, providing a theorem on the limiting distribution of the scaled logarithm of the process.

Contribution

It introduces a new theorem describing the asymptotic behavior of the process's logarithm in critical branching processes with random environments.

Findings

Limiting distribution of scaled log process derived

Conditions for long-term family survival established

Asymptotic behavior characterized as n approaches infinity

Abstract

A critical branching process $\left\{Z_{k},k=0,1,2,...\right\} $ in a random environment generated by a sequence of independent and identically distributed random reproduction laws is considered.\ Let $Z_{p,n}$ be the number of particles at time $p\leq n$ having a positive offspring number at time $n$. \ A theorem is proved describing the limiting behavior, as $% n\rightarrow \infty $ of the distribution of a properly scaled process $\log Z_{p,n}$ under the assumptions $Z_{n}>0$ and $p\ll n$.