Renewal Population Dynamics and their Eternal Family Trees

Abstract

Based on a simple object, an i.i.d. sequence of positive integervalued 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 .Only assuming E[a 0 ] < ∞ and P[a 0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regenerative epochs. 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 regenerative epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regenerative integers form stationary and mixing point processes on Z.

Doeblin trees

“…Another occurrence of a similar object to the bridge graph may be found in [3] in the very special case of integer-valued renewal processes. The dynamics there are slightly different, where instead of specifying a whole process started from each time, one marks each time with the time of death of an individual who is born at that time.…”

“…The dynamics there are slightly different, where instead of specifying a whole process started from each time, one marks each time with the time of death of an individual who is born at that time. This is akin to marking each t ∈ Z by the return time τ (t,x * ) (x * ) of F (t,x * ) to x * , though in [3] these times of death are assumed to be i.i.d., whereas in the present work they have intricate dependence due to the Doeblin-type coupling. The population process defined in [3] is then similar in nature to the sequence of cardinalities of (B t ) t∈Z as considered in Section 4.3.2.…”

“…This is akin to marking each t ∈ Z by the return time τ (t,x * ) (x * ) of F (t,x * ) to x * , though in [3] these times of death are assumed to be i.i.d., whereas in the present work they have intricate dependence due to the Doeblin-type coupling. The population process defined in [3] is then similar in nature to the sequence of cardinalities of (B t ) t∈Z as considered in Section 4.3.2. It is proved in [3] that, under natural conditions, the population process is a stationary regenerative process with independent cycles.…”

“…The population process defined in [3] is then similar in nature to the sequence of cardinalities of (B t ) t∈Z as considered in Section 4.3.2. It is proved in [3] that, under natural conditions, the population process is a stationary regenerative process with independent cycles. In the present work, the process (B t ) t∈Z was shown in Proposition 4.19 to be an irreducible, aperiodic, and positive recurrent Markov chain under suitable conditions, which therefore also admits an i.i.d.…”

“…In order to generalize this concept for infinite networks, instead of picking the root uniformly at random, the network is required to satisfy a mass transport principle. The primary new object of study is the subgraph of bridges between a fixed recurrent state, which is referred to as the bridge graph and is roughly inspired by the population process in [3]. The subgraph is defined by looking at processes started at any time from this fixed state.…”