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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