On the maximum of a type of random processes

TL;DR

This paper studies a class of random processes satisfying a conditional increment condition, providing tail probability estimates and inequalities for their maximum, showing the maximum's tail behavior mirrors that of the process's marginals.

Contribution

It introduces tail probability bounds and Doob-type inequalities for processes with the conditional increment property, linking maximum behavior to marginal distributions.

Findings

Tail probability decay of the maximum matches that of the marginals.

Established a Doob-type inequality for these processes.

Provided estimates for the tail probability of the process maximum.

Abstract

We consider a type of random processes which satisfies the conditional increment condition and obtain an estimate for the tail probability and a Doob-type inequality of the maximum of the process. The main result is that, for processes satisfying the conditional increment condition, the tail probability decay of its maximum behaves in the same way as its marginals.