Abstract:
Let X1, X2, ... be i.i.d. lattice random variables with an irrational span α and let Sn =X1+...+Xn (mod 1). We show that the asymptotic properties of the random walk {Sn, 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 Sn as the Diophantine rank of α passes through a certain critical value.