Large and moderate deviations of weak record numbers in random walks

TL;DR

This paper establishes the asymptotic probabilities of large and moderate deviations for weak record numbers in various types of random walks, enhancing understanding of their deviation behaviors.

Contribution

It provides the first rigorous proof of deviation principles for weak record counts in both right and left continuous random walks.

Findings

Derived asymptotic probabilities for large deviations.

Established moderate deviation principles.

Extended results to different types of continuous random walks.

Abstract

Record numbers are basic statistics in random walks, whose deviation principles are not very clear so far. In this paper, the asymptotic probabilities of large and moderate deviations for numbers of weak records in right continuous or left continuous random walks are proved.