Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912621817495552 |
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| author | Wang, Hua-Ming Zhang, Shuxiong |
| author_facet | Wang, Hua-Ming Zhang, Shuxiong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_19404 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes Wang, Hua-Ming Zhang, Shuxiong Probability 60F05, 60J65, 60J80, 40B05 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. |
| title | Beyond Poisson Approximation: Sums of Markovian Bernoulli Variables with Applications to Brownian Motions and Branching Processes |
| topic | Probability 60F05, 60J65, 60J80, 40B05 |
| url | https://arxiv.org/abs/2504.19404 |