Tightness for processes with fixed points of discontinuities and applications in varying environment

TL;DR

This paper introduces a new sufficient condition for the tightness of stochastic processes, especially those with discontinuities, facilitating analysis in varying environments such as branching processes and processes with catastrophes.

Contribution

It provides a novel tightness criterion applicable to processes with fixed discontinuities, advancing the study of processes in changing or random environments.

Findings

Established a sufficient condition for process tightness with discontinuities

Applied the condition to Galton Watson branching processes in varying environments

Analyzed logistic branching processes with catastrophes using the new criterion

Abstract

We establish a sufficient condition for the tightness of a sequence of stochastic processes. Our condition makes it possible to study processes with accumulations of fixed times of discontinuity. Our motivation comes from the study of processes in varying or random environment. We demonstrate the usefulness of our condition on two examples: Galton Watson branching processes in varying environment and logistic branching processes with catastrophes.