On the local time of random walks associated with Gegenbauer polynomials

TL;DR

This paper investigates the local time behavior of random walks linked to Gegenbauer polynomials in the recurrent case, revealing a Mittag-Leffler distribution as the limit and extending results to certain birth-death Markov chains.

Contribution

It provides the first analysis of local times for Gegenbauer polynomial-based random walks, including explicit limit distributions and a local limit theorem.

Findings

Limit distribution is Mittag-Leffler for nonzero alpha.

Established a local limit theorem for Gegenbauer-based random walks.

Derived limit distributions for specific birth-death Markov chains.

Abstract

The local time of random walks associated with Gegenbauer polynomials $P_n^{(\alpha)}(x),\ x\in [-1,1]$ is studied in the recurrent case: $\alpha\in\ [-\frac{1}{2},0]$. When $\alpha$ is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth and death Markov chains on $\bbN$.