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 ρn of the number of steps needed for a walker starting at the origin to land on the positive semi-axes 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 ρn ∼ 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 deviations from normal scaling are present for the first-passage-time distribution on a semi plane. For more than a century (see, for instance [1]) random walks have played a crucial role as a theoretical tool to model an impressive number of physical (and not only) problems. Fundamental questions in the theory of stochastic processes are related to the problem of when a variable in a system enters some a priori specified state for the first time: the first-passage-time. Knowledge of the first-passage-time distribution (FPTD) finds application in many diverse areas of the natural sciences and economics; from the spike distribution in neuronal dynamics, the meeting time of two molecules in diffusion-controlled chemical reactions, the cluster density in aggregation reactions, to the price of a stock reaching a specific value and the ruin problem of actuarial science (see [2] for an extensive treatment of the problem, and early references to applications); in the last decades, relevance of such a theory to non-equilibrium problems has been exploited [3]. In this framework the Sparre-Andersen theorem (SA) [4][5][6] plays an outstanding role: in physics it has been invoked in the study of persistence in stochastic spin models and random walks [7] the study of polymer dynamics [8] and in the analysis of scattering from a Lorentz slab [9]. In particular, SA states that the probability that a random walker who starts at the origin, enters the positive semi-axis for the first time (its first-passage-time) after n steps is independent of the particular details of the jump length distribution, provided that it is symmetric about the origin and continuous: in such a case the decay of the FPTD has the universal asymptotics n −3/2 . Here the conceptual import of SA is apparent: it provides an outstanding example of universality in the realm of stochastic processes, with a huge universality class. The origin of such universality is subtler than other examples in probability: to our knowledge it cannot be encompassed by simple renormalization, like the central limit theorem (see, fo...
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