Limit law for some modified ergodic sums

An example due to Erdos and Fortet shows that, for a lacunary sequence of integers (q_n) and a trigonometric polynomial f, the asymptotic distribution of normalized sums of f(q_k x) can be a mixture of gaussian laws. Here we give a generalization of their example interpreted as the limiting behavior of some modified ergodic sums in the framework of dynamical systems

Quantitative ergodicity for some switched dynamical systems

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd \times E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology

Limit directions of a vector cocycle, remarks and examples

International audienceWe study the set ${\cal D}(\Phi)$ of limit directions of a vector cocycle $(\Phi_n)$ over a dynamical system, i.e., the set of limit values of $\Phi_n(x) /\|\Phi_n(x)\|$ along subsequences such that $\|\Phi_n(x)\|$ tends to $\infty$. This notion is natural in geometrical models of dynamical systems where the phase space is fibred over a basis with fibers isomorphic to $\mathbb{R}^d$, like systems associated to the billiard in the plane with periodic obstacles. It has a meaning for transient or recurrent cocycles. Our aim is to present some results in a general context as well as for specific models for which the set of limit directions can be described. In particular we study the related question of sojourn in cones of the cocycle when the invariance principle is satisfied

On the CLT for rotations and BV functions

Let $x \mapsto x+ \alpha$ be a rotation on the circle and let $\varphi$ be a step function. We denote by $\varphi\_n (x)$ the corresponding ergodic sums $\sum\_{j=0}^{n-1} \varphi(x+j \alpha)$. Under an assumption on $\alpha$, for example when $\alpha$ has bounded partial quotients, and a Diophantine condition on the discontinuity points of $\varphi$, we show that $\varphi\_n/\|\varphi\_n\|\_2$ is asymptotically Gaussian for $n$ in a set of density 1. The method is based on decorrelation inequalities for the ergodic sums taken at times $q\_k$, where the $q\_k$'s are the denominators of $\alpha$

Central limit theorem for stationary products of toral automorphisms

International audienceLet (A(n)(omega)) be a stationary process in M-d*(Z). For a Holder function f on T-d we consider the sums Sigma(n)(k=1) f((t)A(k)(omega) (t)A(k-1)(omega) ... tA(1)(omega) x mod 1) and prove a Central Limit Theorem for a.e. omega in different situations in particular for "kicked" stationary processes. We use the method of multiplicative systems of Komlos and the Multiplicative Ergodic Theorem

Central limit theorem for products of toral automorphisms

Let $(\tau_n)$ be a sequence of toral automorphisms \tau_n : x \rightarrow A_n x \hbox{ mod }\ZZ^d with $A_n \in {\cal A}$, where ${\cal A}$ is a finite set of matrices in $SL(d, \mathbb{Z})$. Under some conditions the method of "multiplicative systems" of Komlòs can be used to prove a Central Limit Theorem for the sums $\sum_{k=1}^n f(\tau_k \circ \tau_{k-1} \cdots \circ \tau_1 x)$ if $f$ is a Hölder function on $\mathbb{T}^d$. These conditions hold for $2\times 2$ matrices with positive coefficients. In dimension $d$ they can be applied when $A_n= A_n(\omega)$, with independent choices of $A_n(\omega)$ in a finite set of matrices $\in SL(d, \mathbb{Z})$, in order to prove a "quenched" CLT