Reduced critical branching processes in non-favorable random environment

TL;DR

This paper studies the asymptotic behavior of critical branching processes in random environments, revealing how the logarithm of the process scales and converges to a Brownian meander, including cases with stable law increments.

Contribution

It extends existing results by describing the distribution of scaled logarithms of the process conditioned on survival and explores stable law cases.

Findings

Logarithm of the process converges to a Brownian meander under certain conditions.

Distribution of scaled log-process conditioned on survival is characterized.

Results include cases where increments follow a stable law domain of attraction.

Abstract

Let $\left\{ Z_{n},n=0,1,2,...\right\} $ be a critical branching process in i.i.d. random environment, $Z_{r,n}$ be the number of particles in the process at moment $0\leq r\leq n-1$ that have a positive number of descendants in generation $n$, and $\left\{ S_{n},n=0,1,2,...\right\} $ be the associated random walk of $\left\{ Z_{n},n=0,1,2,...\right\} $. It is known that if the increments of the associated random walk have zero mean and finite variance $\sigma ^{2}$ then, for any $t\in \lbrack 0,1]$ \begin{equation*} \lim_{n\rightarrow \infty }\mathbf{P}\left( \frac{\log Z_{\left[ nt\right] ,n}}{\sigma \sqrt{n}}\leq x\Big|Z_{n}>0\right) =\mathbf{P}\left( \min_{t\leq s\leq 1}B_{s}^{+}\leq x\right) ,\;x\in \lbrack 0,\infty ), \end{equation*} where $\left\{ B_{t}^{+},0\leq t\leq 1\right\} $ is the Brownian meander. We supplement this result by description of the distribution of the properly scaled random variable $\log Z_{r,n}$ under the condition $\left\{ S_{n}\leq t\sqrt{k},Z_{n}>0\right\} ,$ where $t>0$ and $r,k\rightarrow \infty $ in such a way that $k=o(n)$ as $n\to\infty$. The case when the distribution of the increments of the associated random walk belongs to the domain of attraction of a stable law is also considered.