Laws Relating Runs, Long Runs, and Steps in Gambler’s Ruin, with Persistence in Two Strata

Abstract

Define a certain gambler’s ruin process $\mathbf {X}_{j}, \, j\ge 0,$ such that the increments $\varepsilon _{j}:=\mathbf {X}_{j}-\mathbf {X}_{j-1}$ take values $\pm 1$ and satisfy $P(\varepsilon _{j+1}=1|\varepsilon _{j}=1, |\mathbf {X}_{j}|=k)=P(\varepsilon _{j+1}=-1|\varepsilon _{j}=-1,|\mathbf {X}_{j}|=k)=a_k$, all $j\ge 1$, where $a_k=a$ if $0\le k\le f-1$, and $a_k=b$ if $f\le k<N$. Here, $0<a, b <1$ denote persistence parameters and $f ,N\in \mathbb {N}$ with $f<N$. The process starts at $\mathbf {X}_0=m\in (-N,N)$ and terminates when $|\mathbf {X}_j|=N$. Denote by ${\mathscr {R}}'_N$, ${\mathscr {U}}'_N$, and ${\mathscr {L}}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler’s ruin process. Define $X_N:=\left( {\mathscr {L}}'_N-\frac{1-a-b}{(1-a)(1-b)}{\mathscr {R}}'_N-\frac{1}{(1-a)(1-b)}{\mathscr {U}}'_N\right) /N$ and let $f\sim \eta N$ for some $0<\eta <1$. We show $\lim _{N\rightarrow \infty } E\{e^{itX_N}\}=\hat{\varphi }(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler’s ruin.