Revisiting John Lamperti's maximal branching process

TL;DR

This paper revisits Lamperti's maximal branching process, analyzing the shape of invariant measures, introducing a truncated chain, and applying hitting time theory to understand its behavior in different regimes.

Contribution

It provides a detailed description of invariant measures and introduces a truncated chain that preserves monotonicity, applying Brown theory and quasi-stationary analysis.

Findings

Characterization of invariant measures in recurrent and transient regimes

Construction of a truncated chain maintaining monotonicity

Application of hitting time theory to finite state-space chain

Abstract

Lamperti's maximal branching process is revisited, with emphasis on the description of the shape of the invariant measures in both the recurrent and transient regimes. A truncated version of this chain is exhibited, preserving the monotonicity of the original Lamperti chain supported by the integers. The Brown theory of hitting times applies to the latter chain with finite state-space, including sharp strong time to stationarity. Additional information on these hitting time problems are drawn from the quasi-stationary point of view.