A Stable Limit Law for Recurrence Times of the Simple Random Walk on the Lattice Z2
- We consider the random walk of a particle on the two-dimensional integer lattice starting at the origin and moving from each site (independently of the previous moves) with equal probabilities to any of the 4 nearest neighbours. When τi denotes the even number of steps between the (i-1)-st and i-th return to the origin, we shall prove that the geometric mean of τ1,...,τn divided by npi converges in distribution to some positive random variable having a logarithmic stable law. We also obtain a rate of this convergence and improve an asymptotic estimate of the tail probability of τ1 due to Erdös and Taylor (1960).
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| Author: | Lothar HeinrichGND, Mirjam Appelt |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-24710 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/2471 |
| Series (Serial Number): | Preprints des Instituts für Mathematik der Universität Augsburg (2013-18) |
| Type: | Preprint |
| Language: | English |
| Date of Publication (online): | 2013/09/25 |
| Publishing Institution: | Universität Augsburg |
| Release Date: | 2013/09/25 |
| Tag: | Esseen's inequality; asymptotic expansion; characteristic function; elliptic integral of first kind; first return time; geometric mean; mathematical constants; simple random walk; square lattice |
| GND-Keyword: | Irrfahrtsproblem; Elliptisches Integral; Geometrisches Mittel; Charakteristische Funktion; Asymptotische Abschätzung; Gitter <Mathematik> |
| Institutes: | Mathematisch-Naturwissenschaftlich-Technische Fakultät |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik | |
| Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Stochastik und ihre Anwendungen | |
| Dewey Decimal Classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
| Licence (German): |  Deutsches Urheberrecht |
