Power and exponential moments of the number of visits and related quantities for perturbed random walks

DOI10.1007/S10959-012-0475-7zbMATH Open1318.60048arXiv1111.4159OpenAlexW2055823445MaRDI QIDQ2346971

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Publication date: 26 May 2015

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Abstract: Let $(x{i}_1,et{a}_1),(x{i}_2,et{a}_2),\mathrm{...}$ be a sequence of i.i.d. copies of a random vector $(xi,eta)$ taking values in $R^2$, and let $S_n:=x{i}_1+\mathrm{...}+x{i}_n$. The sequence $(S_{n-1}+et{a}_n{)}_{ngeq1}$ is then called perturbed random walk. We study random quantities defined in terms of the perturbed random walk: $au(x)$, the first time the perturbed random walk exits the interval $(-infty,x]$, $N(x)$, the number of visits to the interval $(-infty,x]$, and $ho(x)$, the last time the perturbed random walk visits the interval $(-infty,x]$. We provide criteria for the a.s. finiteness and for the finiteness of exponential moments of these quantities. Further, we provide criteria for the finiteness of power moments of $N(x)$ and $ho(x)$.

Full work available at URL: https://arxiv.org/abs/1111.4159

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