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Curve crossing for random walks reflected at their maximum. (English) Zbl 1133.60022

Let \(\{S_n\}\) be a random walk and let \(R_n = \max_{1\leq n} S_j- S_n\) be the random walk reflected in its maximum. The main object of the paper is \(\tau_\kappa(r)= \min\{n\geq 1: R_n> rn^\kappa\}\), i.e., the first passage time above a power law boundary. Here \(\kappa\), \(r> 0\) or \(\kappa= 0\), \(r\geq 0\). The results concern, e.g., a.s.-finiteness of \(\tau_\kappa(r)\), existence of moments of the same as well as growth rate results of \(R_n\) itself.

MSC:

60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems
60K05 Renewal theory
60G40 Stopping times; optimal stopping problems; gambling theory
60F05 Central limit and other weak theorems
60G42 Martingales with discrete parameter
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References:

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