On total progeny of multitype Galton-Watson process and the first passage time of random walk with bounded jumps

TL;DR

This paper develops a method to compute the probability generating function of the total progeny in multitype branching processes and applies it to analyze the first passage times of certain bounded jump random walks.

Contribution

It introduces a novel approach for calculating generating functions of total progeny in multitype branching processes and applies it to specific random walk models.

Findings

Derived probability generating functions for total progeny.

Obtained distributions of hitting times for selected random walks.

Provided explicit formulas for multitype branching processes within random walks.

Abstract

In this paper, we first form a method to calculate the probability generating function of the total progeny of multitype branching process. As examples, we calculate probability generating function of the total progeny of the multitype branching processes within random walk which could stay at its position and (2-1) random walk. Consequently, we could give the probability generating functions and the distributions of the hitting time of corresponding random walks.