title: Sparre-Andersen theorem with spatiotemporal correlations.
abstract: The Sparre-Andersen theorem is a remarkable result in one-dimensional random walk theory concerning the universality of the ubiquitous first-passage-time distribution. It states that the probability distribution ρ(n) of the number of steps needed for a walker starting at the origin to land on the positive semiaxes does not depend on the details of the distribution for the jumps of the walker, provided this distribution is symmetric and continuous, where in particular ρ(n) ∼ n(-3/2) for large number of steps n. On the other hand, there are many physical situations in which the time spent by the walker in doing one step depends on the length of the step and the interest concentrates on the time needed for a return, not on the number of steps. Here we modify the Sparre-Andersen proof to deal with such cases, in rather general situations in which the time variable correlates with the step variable. As an example we present a natural process in 2D that shows that deviations from normal scaling are present for the first-passage-time distribution on a semiplane.
authors.affiliation: Center for Nonlinear and Complex Systems, Università dell' Insubria, Via Valleggio 11, Como 22100, Italy and I.N.F.N. Sezione di Milano, Via Celoria 16, Milano 20133, Italy.
authors.forename: Roberto
authors.initials: R
authors.lastname: Artuso
authors.affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy.
authors.forename: Giampaolo
authors.initials: G
authors.lastname: Cristadoro
authors.affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy.
authors.forename: Mirko Degli
authors.initials: MD
authors.lastname: Esposti
authors.affiliation: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy.
authors.forename: Georgie
authors.initials: G
authors.lastname: Knight
mesh_terms: D000818:Animals; D003198:Computer Simulation; D006801:Humans; D008954:Models, Biological; D008956:Models, Chemical; D008958:Models, Molecular; D015233:Models, Statistical; D062211:Spatio-Temporal Analysis