arXiv subjects
Publications and source records attributed to Francoise Pene.
Let T be an ergodic automorphism of the d-dimensional torus. In the spirit of Le Borgne, we give conditions on the Fourier coeffi cients of a real valued function f under which the Birkhoff sums satis fy a strong invariance principle. Next, reinforcing the condition on the Fourier coeffi cients in a natural way, we obtain explicit rates of convergence in the strong invariance principle.
In this paper, we extend a result of Kesten and Spitzer (1979). Let us consider a stationary sequence $(ξ\_k:=f(T^k(.)))\_k$ given by an invertible probability dynamical system and some centered function $f$. Let $(S\_n)\_n$ be a simple symmetric random walk on $Z$ independent of $(ξ\_k)\_k$. We give examples of partially hyperbolic dynamical systems and of functions $f$ such that $n^{-3/4}(ξ(S\_1)+...+ξ(S\_k))$ converges in distribution as $n$ goes to infinity.
The motivation of this work is the study of the error term e_t^ε(x,ω) in the averaging method for differential equations perturbed by a dynamical system. Results of convergence in distribution for (\frac{e_t^ε(x,\cdot)}{\sqrtε})_{ε>0} have been established in Khas'minskii [Theory Probab. Appl. 11 (1966) 211-228], Kifer [Ergodic Theory Dynamical Systems 15 (1995) 1143-1172] and Pène [ESAIM Probab. Statist. 6 (2002) 33-88]. We are interested here in the question of the rate of convergence in distribution of the family of random variables (\frac{e_t^ε(x,\cdot)}{\sqrtε})_{ε>0} when εgoes to 0 (t>0 and x\inR^d being fixed). We will make an assumption of multiple decorrelation property (satisfied in several situations). We start by establishing a simpler result: the rate of convergence in the central limit theorem for regular multidimensional functions. In this context, we prove a result of convergence in distribution with rate of convergence in O(n^{-1/2+α}) for all α>0 (for the Prokhorov metric). This result can be seen as an extension of the main result of Pène [Comm. Math. Phys. 225 (2002) 91-119] to the case of d-dimensional functions. In a second time, we use the same method to establish a result of convergence in distribution for (\frac{e_t^ε(x,\cdot)}{\sqrtε})_{ε>0} with rate of convergence in O(ε^{1/2-α}) (for the Prokhorov metric).
We prove a central limit theorem with speed $n^{-1/2}$ for stationary processes satisfying a strong decorrelation hypothesis. The proof is a modification of the proof of a theorem of Rio. It is elementary but quite long and technical.