Renewal processes, population dynamics, and unimodular trees

Abstract

Based on a simple object, an i.i.d. sequence of positive integer-valued random variables {a_n}_n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + a_n. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {a_n}_n∊ℤ. Second, the directed random graph T^f on ℤ generated by the random map f(n) = n + a_n. Assuming only that E [a₀] < ∞ and P [a₀ = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, T^f is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, T^f splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.