Asymptotic behaviour near extinction of continuous state branching processes

TL;DR

This paper investigates the near-extinction behavior of sub-critical continuous state branching processes, establishing a law of the iterated logarithm analogue for these processes and their reflected counterparts.

Contribution

It introduces a new asymptotic law near extinction for continuous state branching processes, extending classical results to this stochastic process class.

Findings

Established an analogue of Khintchin's law of the iterated logarithm near extinction.

Derived asymptotic behavior for the reflected process at its infimum.

Provided conditions under which the law applies.

Abstract

In this note, we study the asymptotic behaviour near extinction of (sub-) critical continuous state branching processes. In particular, we establish an analogue of Khintchin's law of the iterated logarithm near extinction time for a continuous state branching process whose branching mechanism satisfies a given condition and its reflected process at its infimum.