Trigonometric polynomials and multidimensional random walks: application to the ergodic theory

Let {S-k, k greater than or equal to 0} be a random walk define on the probability space (Ohm, B, P) as the partial sum of the k first terms of a sequence of I.I.D. random vectors Z(d) (d greater than or equal to 1) valued, with delta (delta > 0) positive moment. Given any probability measure nu on the torus T-d We give an upper bound for parallel to1/N Sigma (N-1)(k=0) exp 2i pi [alpha, S-k] parallel to (2,nu>(*) over bar * (d alpha)). We use random methods based on decoupling inequalities fur random Gaussian processes. We apply these results in ergodic theory: given any dynamical system (X, A, mu, T) such that T mu = mu, for f is an element of L-2(mu) we give conditions on the spectral measure of T to obtain mu -almost sure convergence of the ergodic average {1/N Sigma (N)(k=1) f o T-Sk(omega), N greater than or equal to 1} universally for the random walk. (C) 2000 Editions scientifiques et medicales Elsevier SAS.