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 $\varepsilon_{j}:=\mathbf{X}_{j}\mathbf{X}_{j1}$ take values $\pm1$ 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 f1$, 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 ${\cal R}'_N$, ${\cal U}'_N$, and ${\cal L}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left ({\cal L}'_N\frac{1ab}{(1a)(1b)}{\cal R}'_N\frac{1}{(1a)(1b)}{\cal U}'_N\right )/N$ and let $f\sim\eta N$ for some $0<\eta <1$. We show $\lim_{N\to\infty} E\{e^{itX_N}\}=\hat{\varphi}(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler's ruin.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.08468
 Bibcode:
 2017arXiv171008468M
 Keywords:

 Mathematics  Probability;
 60F05
 EPrint:
 Presented 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