Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata (1710.08468v3)

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. We obtain a companion theorem for the last visit portion of the gambler's ruin.