arXiv subjects

A. Shklyaev

Publications and source records attributed to A. Shklyaev.

2 recordsLinked to original sources

arXiv2026

Branching Process in a Varying Environment: How to Grow Like the Product of Means

Consider a branching process $\{Z_n\}$ in a varying environment. Let $\{W_n\}$ be the natural martingale $Z_n/{\bf E}Z_n$. It converges to some random variable $W$ as $n\to\infty$. An important problem is to show that ${\bf P}(W>0)$ equals the survival probability, so that $Z_n$ is either $0$ or of the order ${\bf E}Z_n$. We find a new kind of sufficient conditions, applicable to branching processes in a random environment. An important property of our estimates is that we don't necessary assume that ${\bf E}X_{i,1}^2$ are finite for every $i$.

math.PR↗

arXiv2025

On the Asymptotics of the Connectivity Probability of Erdos-Renyi Graphs

In this paper, we investigate the exact asymptotic behavior of the connectivity probability in the Erdos-Renyi graph G(n,p), under different asymptotic assumptions on the edge probability p=p(n). We propose a novel approach based on the analysis of inhomogeneous random walks to derive this probability. We show that the problem of graph connectivity can be reduced to determining the probability that an inhomogeneous random walk with Poisson-distributed increments, conditioned to form a bridge, is actually an excursion

math.PR↗