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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