An Arcsine Law for Markov Random Walks

Abstract

The classic arcsine law for the number $N_{n}^{>}:=n^{-1}\sum_{k=1}^{n}\mathbf{1}_{\{S_{k}>0\}}$ of positive terms, as $n\to\infty$, in an ordinary random walk $(S_{n})_{n\ge 0}$ is extended to the case when this random walk is governed by a positive recurrent Markov chain $(M_{n})_{n\ge 0}$ on a countable state space $\mathcal{S}$, that is, for a Markov random walk $(M_{n},S_{n})_{n\ge 0}$ with positive recurrent discrete driving chain. More precisely, it is shown that $n^{-1}N_{n}^{>}$ converges in distribution to a generalized arcsine law with parameter $\rho\in [0,1]$ (the classic arcsine law if $\rho=1/2$) iff the Spitzer condition $$ \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\mathbb{P}_{i}(S_{n}>0)\ =\ \rho $$ holds true for some and then all $i\in\mathcal{S}$, where $\mathbb{P}_{i}:=\mathbb{P}(\cdot|M_{0}=i)$ for $i\in\mathcal{S}$. It is also proved, under an extra assumption on the driving chain if $0<\rho<1$, that this condition is equivalent to the stronger variant $$ \lim_{n\to\infty}\mathbb{P}_{i}(S_{n}>0)\ =\ \rho. $$ For an ordinary random walk, this was shown by Doney for $0<\rho<1$ and by Bertoin and Doney for $\rho\in\{0,1\}$.