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

Research paper by Gregory J. Morrow

Indexed on: 23 Oct '17Published on: 23 Oct '17Published in: arXiv - Mathematics - Probability



Abstract

Define a certain gambler's ruin process $\mathbf{X}_{j}, \mbox{ \ }j\ge 0,$ such that the increments $\varepsilon_{j}:=\mathbf{X}_{j}-\mathbf{X}_{j-1}$ 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 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 ${\cal R}'_N$, ${\cal U}'_N$, and ${\cal L}'_N$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler's ruin process. Define $X_N:=\left ({\cal L}'_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}'_N-\frac{1}{(1-a)(1-b)}{\cal U}'_N\right )/N$ 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. In case $b=1-a$ and $\eta=a$, $\hat{\varphi}(t)=\sigma^2 t/\{\sinh(\sigma t)[\sigma \cosh(\sigma t)+i (1-2a)^2\sinh(\sigma t)]\}$, for $\sigma^2:=1-3a+3a^2$. If $b=a$, then $\hat{\varphi}(t)=At/\sinh(A t)$, for $A:=\sqrt{a/(1-a)}$.