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

Let \ηᵢ\₈ ₁ be a sequence of dependent Bernoulli random variables. While the Poisson approximation for the distribution of ₈=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 ₈=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.