### G-2019-23

# Detecting periodicity from the trajectory of a random walk in random environment

## Bruno Rémillard and Jean Vaillancourt

For nearest neighbor univariate random walks in a periodic environment, where the probability of moving depends on a periodic function, we show how to estimate the period and the function. For random walks in non-periodic environments, we find that the asymptotic limit of the estimator is constant in the ballistic case, when the random walk is transient and the law of large numbers holds with a non zero limit. Numerical examples are given in the recurrent case, and the sub-ballistic case, where the random walk is transient but the law of large numbers yields a zero limit.

Published **March 2019**
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15 pages