Limit theorems for signatures

We obtain strong moment invariance principles for normalized multiple iterated sums and integrals of the form (Formula Presented) are centered stationary vector processes with some weak dependence properties. We show in both cases that the process S^(ν) N is O(N^−δ ), δ > 0 is close in the moment variational norm to a certain stochastic process W^(ν) N constructed recursively starting from WN = W(1) N which is a Brownian motion with covariances. This implies the similar estimate for the distance in the Prokhorov and the Wasserstein metrics between distributios of the above processes. This is done by constructing a coupling between S⁽¹⁾ N and W(1) N, estimating directly the moment variational norm of S^(ν) N^−W(ν)N for ν = 1, 2 and extending these estimates to ν > 2 relying on arguments borrowed from the rough paths theory. In the continuous time we work both under direct weak dependence assumptions and also within the suspension setup which is more appropriate for applications in dynamical systems. In Appendix we derive large deviations estimates for iterated sums and integrals.