Defective Galton-Watson processes

2016/12/12 by Serik Sagitov, Sagitov, Serik, Carmen Minuesa +1
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1612.03588

openalex publication_date 2016/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Galton-Watson process is a Markov chain modeling the population size of independently reproducing particles giving birth to k offspring with probability p_k, k≥0. In this paper we consider \it defective Galton-Watson processes having defective reproduction laws, so that ∑_k≥0p_k=1-\eps for some \eps∈(0,1). In this setting, each particle may send the process to a graveyard state Δ with probability \eps. Such a Markov chain, having an enhanced state space \0,1,…\∪\Δ\, gets eventually absorbed either at 0 or at Δ. Assuming that the process has avoided absorption until the observation time t, we are interested in its trajectories as t→∞ and \eps→0.

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