Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata

Abstract

Define a certain gambler's ruin process $\mathbf{X}_{j}, \mbox{ \ }j\ge 0,$ such that the increments ${\epsilon }_j:={\mathbf{X}}_j-{\mathbf{X}}_{j-1}$ take values $\pm 1$ and satisfy $P({\epsilon }_{j+1}=1|{\epsilon }_j=1,|{\mathbf{X}}_j|=k)=P({\epsilon }_{j+1}=-1|{\epsilon }_j=-1,|{\mathbf{X}}_j|=k)=a_k$, 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 ${\mathcal{R}}_N^{\prime }$, ${\mathcal{U}}_N^{\prime }$, and ${\mathcal{L}}_N^{\prime }$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left({\mathcal{L}}_N^{\prime }-\frac{1-a-b}{(1-a)(1-b)}{\mathcal{R}}_N^{\prime }-\frac{1}{(1-a)(1-b)}{\mathcal{U}}_N^{\prime }\right)/N$ and let $f\sim \eta N$ for some $0<\eta <1$. We show ${\lim }_{N\to \infty }E\{e^{it{X}_N}\}=\hat{\phi }(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler's ruin.

arXiv commentPresented at 8th International Conference on Lattice Path Combinatorics, Cal Poly Pomona, Aug., 2015. The 2nd version has been streamlined, with references added, including reference to a companion document with details of calculations via Mathematica. The 3rd version has 2 new figures and improved presentation