Recurrence of stationary random walks

If (X_n) is a (real-valued) stationary process, then the associated (one-dimensional) random walk is sequence of random variables (Y_n=X_1+\dots +X_n, n\ge 1). The random walk (Y_n) is recurrent if it comes back to zero (or at least arbitrarily close to zero) with probability one.

Classical probability theory deals with the case where the random variables (X_n) are independent and characterizes recurrence of (Y_n) in terms of the distribution of X_1.

By using methods from ergodic theory one can remove the hypothesis of independence and obtain very general conditions for recurrence of (Y_n) of arbitrary stationary random walks in one or more dimensions.

We shall also discuss some of the mysteries associated with recurrence and transience (i.e. non-recurrence) of stationary and nonstationary random walks.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).