Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes

2025/04/28 by H Wang, Wang, Hua-Ming, Shuxiong Zhang +1
Mathematics · #40B05 #60F05 #60J65 #60J80 #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2504.19404

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

Abstract

Let \η_i\_{i≥ 1} be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of ∑_i=1ⁿη_i has been extensively studied in the literature, this paper establishes new convergence regimes characterized by non-Poisson limits. Specifically, under a Markovian dependence structure, we show that ∑_i=1ⁿη_i, under suitable scaling, converges almost surely or in distribution as n→∞ to a geometric or Gamma random variable. These results provide a new tool for analyzing the limit distributions of sums of Markovian dependent Bernoulli random variables. We demonstrate these results in several applications: determining the limiting distribution of the number of weak cutspheres for a d(≥3)-dimensional standard Brownian motion; deriving the limit law for weak cutpoints of geometric Brownian motion; and analyzing how often the population size reaches a given threshold in certain branching processes, both with and without immigration.

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