Favorite Downcrossing Sites of One-Dimensional Simple Random Walk

TL;DR

This paper investigates the properties of favorite downcrossing sites in a one-dimensional simple symmetric random walk, revealing finiteness and recurrence properties of these sites over time.

Contribution

It provides new results on the cardinality and frequency of favorite downcrossing sites, including finiteness of certain configurations and recurrence of others.

Findings

Finitely many times with at least four favorite downcrossing sites.

Three favorite downcrossing sites occur infinitely often.

Open questions related to the behavior of these sites are discussed.

Abstract

Random walk is a very important Markov process and has important applications in many fields.For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local time at time $n$ achieves the maximum among all sites. In this paper, we study the cardinality of the favorite downcrossing site set, and will show that with probability 1 there are only finitely many times at which there are at least four favorite downcrossing sites and three favorite downcrossing sites occurs infinitely often. Some related open questions will be introduced.