Almost sure invariance principle for dynamical systems by spectral methods

Abstract

25 pages v2: minor revision v3: published versionInternational audienceWe prove the almost sure invariance principle for stationary R^d--valued processes (with dimension-independent very precise error terms), solely under a strong assumption on the characteristic functions of these processes. This assumption is easy to check for large classes of dynamical systems or Markov chains, using strong or weak spectral perturbation arguments

On the rate of convergence in the weak invariance principle for dependent random variables with applications to Markov chains

“…The results of the paper can be also applied to a large class of dynamical systems, however this stays beyond the scope of the present paper. For a discussion of these type of applications we refer to [15].…”

“…Moreover, we give explicit expressions of some constants involved in the rate of convergence; for instance, in the case of Markov walks we are able to figure out the dependence of the rate of convergence on the properties of the Banach space related to the corresponding family of perturbed transition operators (P t ) |t|≤ε 0 and on the initial state X 0 = x of the associated Markov chain. When compared with the rate N − 1 2 α 1+2α in the almost sure invariance principle of [15] ours appears with a loss in the power of multiple 2+2α 3+2α < 1. This loss in the power is exactly the same as in the case of independent r.v.…”

“…Examples are Markov chains, whose perturbed transition probability operators (P t ) |t|≤ε 0 exhibit a spectral gap and enough regularity in t, and dynamical systems, whose characteristic functions can be coded by a family of operators (L t ) |t|≤ε 0 satisfying similar properties. Gouëzel proves in [15] an almost sure invariance principle with a rate of convergence. The proof relies on a progressive blocking technique (see Bernstein [2]) coupled with a triadic Cantor like scheme and on the Komlós, Major and Tusnády approximation type results for independent r.v.…”

“…Gouëzel [15] has introduced a new type of mixing condition which is tied to spectral properties of the sequence (X k ) k≥1 . Consider the vectors X 1 = X J 1 , ..., X J M 1 and X 2 = X kgap+J M 1 +1 , ..., X kgap+J M 1 +M 2 , called the past and the future, respectively, where X k+Jm = l∈Jm X k+l , J m = [j m−1 , j m ), j 0 ≤ ... ≤ j M 1 +M 2 , and k gap is a gap between X 1 and X 2 .…”

“…Consider the vectors X 1 = X J 1 , ..., X J M 1 and X 2 = X kgap+J M 1 +1 , ..., X kgap+J M 1 +M 2 , called the past and the future, respectively, where X k+Jm = l∈Jm X k+l , J m = [j m−1 , j m ), j 0 ≤ ... ≤ j M 1 +M 2 , and k gap is a gap between X 1 and X 2 . Roughly speaking, the condition used in [15] suppose that the characteristic function of X 1 , X 2 is exponentially closed to the product of the characteristic functions of the past X 1 and the future X 2 , with an error term of the form A exp (−λk gap ) , where λ is some non negative constant and A is polynomial in terms of the size of the blocks. This mixing property is particularly suited for systems whose behavior can be described in terms of spectral properties of some related family of operators, as initiated by Nagaev [29], [30] and Guivarc'h [16].…”