Conditional central limit theorem for critical branching random walk

2023/11/18 by Wenming Hong, Hong, Wenming, Shengli Liang +1 · 1 citation
Mathematics · Computer Science · #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods #Bayesian Methods and Mixture Models

paper · pdf · doi:10.48550/arxiv.2311.11044

Abstract

Consider a critical branching random walk on ℝ. Let Z⁽ⁿ⁾(A) be the number of individuals in the n-th generation located in A∈ B(ℝ) and Z_n:=Z⁽ⁿ⁾(ℝ) denote the population of the n-th generation. We prove that, under some conditions, for all x∈ ℝ, as n→ ∞, L(\fracZ⁽ⁿ⁾(-∞, √(n) x]n ~ |~ Z_ngt;0) \LongrightarrowL(Y(x)), where ⇒ means weak convergence and Y(x) is a random variable whose distribution is specified by its moments.

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