Estimation of subcritical Galton Watson processes with correlated immigration

2024/04/18 by Yacouba Boubacar Maïnassara, Mainassara, Yacouba Boubacar, Rabehasaina, Landy
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2404.12137

openalex publication_date 2024/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider an observed subcritical Galton Watson process \Y_n, n∈ ℤ \ with correlated stationary immigration process \ε_n, n∈ ℤ \. Two situations are presented. The first one is when Cov(ε₀,ε_k)=0 for k larger than some k₀: a consistent estimator for the reproduction and mean immigration rates is given, and a central limit theorem is proved. The second one is when \ε_n, n∈ ℤ \ has general correlation structure: under mixing assumptions, we exhibit an estimator for the the logarithm of the reproduction rate and we prove that it converges in quadratic mean with explicit speed. In addition, when the mixing coefficients decrease fast enough, we provide and prove a two terms expansion for the estimator. Numerical illustrations are provided.

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