Limit theorems for pure death processes coming down from infinity

Abstract

We consider a pure death process (Z(t), t ≥ 0) with death rates λn satisfying the condition ∞ n=2 λ −1 n < ∞ of coming from infinity, Z(0) = ∞, down to an absorbing state n = 1. We establish limit theorems for Z(t) as t → 0, which strengthen the results that can be extracted from [1]. We also prove a large deviation theorem assuming that λn regularly vary as n → ∞ with an index β > 1. It generalises a similar statement with β = 2 obtained in [4] for λn = n 2 .

Rank-dependent Galton‒Watson processes and their pathwise duals

“…the result is a pure-death process coming down from infinity (see, e.g. [23]). Observe that in this case the dual Markov chain gives a defective reproduction model which is not a rank-dependent GW process.…”

Time-changed spectrally positive Lévy processes starting from infinity

“…A natural further study is to investigate the small-time asymptotics of the process when it leaves an entrance boundary. Such studies have been carried out for instance for the boundary ∞ of birth-death processes, see Bansaye et al [4], Sagitov and France [33], and for Kolmogorov diffusions, see Bansaye et al [3]. We also refer to the work of Bansaye [2] for a general method of comparing stochastic processes with deterministic flows to study the coming down from infinity.…”