Poisson recurrence times

Extract

Let Y₁, Y₂, … be a sequence of independent and identically distributed Poisson random variables with parameter λ. Let S_n = Y₁ + … + Y_n, n = 1,2,…, S₀ = 0. The event S_n = n is a recurrent event in the sense that successive waiting times between recurrences form a sequence of independent and identically distributed random variables. Specifically, the waiting time probabilities are (Alternately, the f_n can be described as the probabilities for first return to the origin of the random walk whose successive steps are Y₁ − 1, Y₂ − 1, ….)