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
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/2471
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2013-18)
Publishing Institution:Universität Augsburg
Release Date:2013/09/25
Tag:simple random walk; square lattice; first return time; geometric mean; characteristic function; elliptic integral of first kind; asymptotic expansion; Esseen's inequality; mathematical constants
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 mit Print on Demand