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

@article{Morrow2019LawsRR,
title={Laws relating runs, long runs, and steps in gambler's ruin, with persistence in two strata},
author={G. Morrow},
journal={arXiv: Probability},
year={2019},
pages={343-381}
}
• G. Morrow
• Published 2019
• Mathematics
• arXiv: Probability
Define a certain gambler’s ruin process $$\mathbf {X}_{j}, \, j\ge 0,$$ such that the increments $$\varepsilon _{j}:=\mathbf {X}_{j}-\mathbf {X}_{j-1}$$ 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… Expand

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