Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Hua-Ming Wang, Shuxiong Zhang

TL;DR

The paper studies the limit laws for sums of dependent Bernoulli variables under Markovian dependence, showing that non-Poisson limits such as Geo and Gamma arise under suitable scaling. It develops a moment-based framework, leveraging asymptotics of multiple index sums and regular variation to characterize regimes where $\sum_{j=1}^n\eta_j$ converges to Geo, Exp, or Gamma distributions. The results are then applied to stochastic processes including Brownian motion (weak cutspheres and cutpoints) and branching processes (Galton–Watson and BPVE with immigration), yielding precise limit distributions for counts of threshold events. The work provides a unified methodology to analyze limit distributions of sums of Markovian Bernoulli variables and offers tools with potential impact in stochastic modeling and probabilistic limit theory.

Abstract

Let $\{η_i\}_{i\ge 1}$ be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of $\sum_{i=1}^nη_i$ 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 $\sum_{i=1}^nη_i,$ under suitable scaling, converges almost surely or in distribution as $n\to\infty$ 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 $d(\ge3)$-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.

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

TL;DR

The paper studies the limit laws for sums of dependent Bernoulli variables under Markovian dependence, showing that non-Poisson limits such as Geo and Gamma arise under suitable scaling. It develops a moment-based framework, leveraging asymptotics of multiple index sums and regular variation to characterize regimes where converges to Geo, Exp, or Gamma distributions. The results are then applied to stochastic processes including Brownian motion (weak cutspheres and cutpoints) and branching processes (Galton–Watson and BPVE with immigration), yielding precise limit distributions for counts of threshold events. The work provides a unified methodology to analyze limit distributions of sums of Markovian Bernoulli variables and offers tools with potential impact in stochastic modeling and probabilistic limit theory.

Abstract

Let be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of 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 under suitable scaling, converges almost surely or in distribution as 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 -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.
Paper Structure (21 sections, 10 theorems, 179 equations)

This paper contains 21 sections, 10 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Background, motivation and notation
  3. Main results
  4. Applications
  5. Weak cutspheres of $d(\ge3)$-dimensional Brownian motions
  6. Weak cutpoints of geometric Brownian motion
  7. Times the population size of a branching process reaches a given threshold
  8. Asymptotic analysis of multiple sums
  9. Proof of Part (i) of Theorem \ref{['prpd']}
  10. Proof of Part (ii) of Theorem \ref{['prpd']}
  11. Proof of Corollary \ref{['rzr']}
  12. Proofs of Theorems \ref{['thbb']} and \ref{['tha']}
  13. Proof of Part (i) of Theorem \ref{['thbb']}
  14. Proof of Part (ii) of Theorem \ref{['thbb']}
  15. Proof of Theorem \ref{['tha']}
  16. ...and 6 more sections

Key Result

Theorem 1

Let $\{\eta_{n}\}_{\ n\ge 1}$ be a sequence of Bernoulli variables such that mc is satisfied. Assume that $\{D(n)\}_{n\ge0}$ is a sequence of numbers such that $D(0)=1$ and $D(n)>1$ for all $n\ge1.$

Theorems & Definitions (19)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Remark 2
  • Definition 2
  • Corollary 1
  • Remark 3
  • Definition 3
  • Corollary 2
  • ...and 9 more