Large and moderate deviations for record numbers in some non-nearest neighbor random walks

TL;DR

This paper investigates the probabilities of large and moderate deviations in record numbers for non-nearest neighbor random walks, providing new asymptotic results and highlighting limitations of traditional Brownian motion analysis methods.

Contribution

It derives the asymptotic deviation probabilities for record counts in non-nearest neighbor random walks, a case not fully explored before.

Findings

Asymptotic probabilities for large deviations established

Analysis depends solely on direct random walk methods

Traditional Brownian motion techniques are inadequate for these cases

Abstract

The deviation principles of record numbers in random walk models have not been completely investigated, especially for the non-nearest neighbor cases. In this paper, we derive the asymptotic probabilities of large and moderate deviations for the number of "weak records"(or "ladder points") in two kinds of one-dimensional non-nearest neighbor random walks. The proofs depend only on the direct analysis of random walks. We illustrate that the traditional method of analyzing the local time of Brownian motions, which is often adopted for the simple random walks, would lead to wrong conjectures for our cases.