Sharp moderate maximal inequalities for upward skip-free Markov chains

TL;DR

This paper develops sharp moderate maximal inequalities for upward skip-free Markov chains, extending classical $L^p$ inequalities for martingales, and applies these results to queueing models revealing phase transition phenomena.

Contribution

It introduces the first sharp moderate maximal inequalities for upward skip-free Markov chains, generalizing $L^p$ inequalities and analyzing specific queueing examples.

Findings

Established sharp moderate maximal inequalities for upward skip-free Markov chains.

Applied inequalities to M/M/1 queue and a large death jumps chain.

Discovered phase transition in the M/M/1 queue but not in the large jumps chain.

Abstract

The $L^p$ maximal inequalities for martingales are one of the classical results in probability theory. Here we establish the sharp moderate maximal inequalities for upward skip-free Markov chains, which include the $L^p$ maximal inequalities as special cases. Furthermore, we apply our theory to two specific examples and obtain their moderate maximal inequalities: the first one is the M/M/1 queue and the second one is an upward skip-free Markov chain with large death jumps. These two examples have the same total birth and death rates. However, the former exhibits a phase transition phenomenon while the latter does not.