• arcsine law
• ergodic theory
• random walk

Given a symmetric random walk on the integers, the proportion of times that the process spends on the positive side is a very natural subject to question about. Paul Levy showed, in 1939, that this proportion, in the limit, is distributed like a random variable with density 1/pi*sqrt(x(1-x)), which is known as the arcsine distribution.
Motivated by this classical result, we consider a random walk driven by a deterministic system. Let T:X \rightarrow X be a continuous mapping in a measure space, and f:X \rightarrow Z a zero-mean function. We are interested in the proportion of times that S_n(x)=\sum_{k=0}^{n-1}f(T^k(x)) is positive as n tends to infinity. To address this, we explore two generalizations of the arcsine law, one probabilistic the other dynamic.
Following Lamperti's work from 1956, the arcsine law is extended to a broader class of discrete-time stochastic processes, including all Markov chains where the state space is divided into two regions by a recurrent state. Lamperti's results led the foundations for all future developments in the area. One significant extension is deeply analized here: referring to Thaler-Zweimuller (2004), we present the arcsine law in the context of infinite-measure dynamical systems. The main result asserts that for a conservative, ergodic, measure-preserving transformation on an infinite-measure space, where the space is separated by a recurrent state in two regions, the mean sojourn time in one of these two subspaces converges weakly to a generalized arcsine law. This convergence holds under certain asymptotic conditions of the relative transfer operator.
This theorem could be applied to the scenario of a deterministic random walk, supporting the idea that fluctuations in such systems are qualitatively similar to those observed for a classical random walk. Finally, we present the results of numerical simulations performed within this framework, with particular emphasis on the one involving the logistic map.