Budapest-Vienna Probability Seminar, István Berkes (Rényi): Random walks on the circle and Diophantine approximation

Abstract:

Let X₁, X₂, ... be i.i.d. lattice random variables with an irrational span α and let S_n =X₁+...+X_n (mod 1). We show that the asymptotic properties of the random walk {S_n, n=1, 2, ...} are closely connected with the rational approximation properties of α and in particular, we point out an interesting critical phenomenon, i.e. a sudden change in the convergence speed in limit theorems for S_n as the Diophantine rank of α passes through a certain critical value.