Recurrence rates and hitting-time distributions for random walks on the line. (English) Zbl 1266.60084
Summary: We consider random walks on the line given by a sequence of independent identically distributed jumps belonging to the strict domain of attraction of a stable distribution, and first determine the almost sure exponential divergence rate, as \(\varepsilon \to 0\), of the return time to \((-\varepsilon,\varepsilon)\). We then refine this result by establishing a limit theorem for the hitting-time distributions of \((x-\varepsilon,x+\varepsilon)\) with arbitrary \(x\in \mathbb{R}\).
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60F05 | Central limit and other weak theorems |
References:
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