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

Define a certain gambler’s ruin process ${\mathbf{X}}_j,j\ge 0,$ such that the increments ${\epsilon }_j:={\mathbf{X}}_j-{\mathbf{X}}_{j-1}$ take values $\pm 1$ 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 $\mathrm{\mid }{\mathbf{X}}_j\mathrm{\mid }=N$. Denote by ${\mathcal{R}}_N^{\mathrm{\prime }}$, ${\mathcal{U}}_N^{\mathrm{\prime }}$, and ${\mathcal{L}}_N^{\mathrm{\prime }}$, respectively, the numbers of runs, long runs, and steps in the meander portion of the gambler’s ruin process. Define $X_N:=\left ({\cal L}‘_N-\frac{1-a-b}{(1-a)(1-b)}{\cal R}’_N-\frac{1}{(1-a)(1-b)}{\cal U}'N\right )/N$andlet$f\sim\eta N$forsome$0<\eta <1$\mathrm{.}Weshow$\lim{N\to\infty} E{e^{itX_N}}=\hat{\varphi}(t)$ exists in an explicit form. We obtain a companion theorem for the last visit portion of the gambler’s ruin.