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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 portionof the gambler's ruin process. Define $X_N:=\left ({\calL}'_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}'_N-\frac{1}{(1-a)(1-b)}{\cal U}'_N\right)/N$ 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 acompanion theorem for the last visit portion of the gambler's ruin.

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