An Arcsine Law for Markov Random Walks

Abstract

The classic arcsine law for the number $N_n^{>}:=n^{-1}{\sum }_{k=1}^n{1}_{\{S_k>0\}}$ 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 $$\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{k=1}^n{\mathbb{P}}_i(S_n>0)=\rho$$ 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 $$\underset{n\to \infty }{\lim }{\mathbb{P}}_i(S_n>0)=\rho .$$ For an ordinary random walk, this was shown by Doney for $0<\rho <1$ and by Bertoin and Doney for $\rho \in \{0,1\}$.