On explicit form of the Kolmogorov constant in the theory of Galton-Watson Branching Processes

TL;DR

This paper derives an explicit formula for the Kolmogorov constant in subcritical Galton-Watson branching processes, linking it to process parameters using asymptotic analysis of generating functions and transition probabilities.

Contribution

It provides a new explicit expression for the Kolmogorov constant based on the structural parameters of the process, advancing theoretical understanding.

Findings

Explicit formula for the Kolmogorov constant derived

Asymptotic properties of transition probabilities analyzed

Connection established between the constant and process parameters

Abstract

The paper considers the well-known Galton-Watson stochastic branching process. We are dealing with a non-critical case. In the subcritical case, when the mean of the direct descendants of one particle per generation of the time step is less than 1, the population mean of the number of particles on the positive trajectories of the process stabilizes and approaches 1/K, where K is the so-called Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the process. Our reasoning is essentially based on the Basic Lemma, which describes the asymptotic expansion of the generating function of the distribution of the number of particles. An important role is also played by the asymptotic properties of the transition probabilities of the so-called Q-process and their property convergence to invariant measures.