One Explicitly Solvable Model For The Galton-Watson Processes In the Random Environment

2020/11/28 by Han Dan, Han, Dan, Stanislav Molchanov +3
Computer Science · Economics, Econometrics and Finance · Mathematics · #60G51 #60J80 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2011.14171

openalex publication_date 2020/11/28 · openalex created_date 2020/12/07 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the Galton-Watson process in the random environment for the particular case when the number of the offsprings in each generation has the fractional linear generation function with random parameters. In this case, the distribution of N_t, the number of particles at the moment time t=0,1,2,⋯ can be calculated explicitly. We present the classification of such processes and limit theorems of two types: quenched type which is for the fixed realization of the random environment and annealed type which includes the averaging over the environment.

Citations

• On Branching Processes with Random Environments: I: Extinction Probabilities
• Limit theorems for supercritical MBPRE with linear fractional offspring distributions

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