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

Let \ηᵢ≥ 1 be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of ∑ᵢ₌₁ⁿηᵢ 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₌₁ⁿηᵢ, 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.