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DCDS

Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$
be a countable group of $\nu$-preserving invertible maps of $X$ into
itself. To a probability measure $\mu$ on $\Gamma$ corresponds a
random walk on $X$ with Markov operator $P$ given by $P\psi(x) =
\sum_{a} \psi(ax) \, \mu(a)$. We consider various examples of
ergodic $\Gamma$-actions and random walks and their extensions by a
vector space: groups of automorphisms or affine transformations on
compact nilmanifolds, random walks in random scenery on non amenable
groups, translations on homogeneous spaces of simple Lie groups,
random walks on motion groups. A powerful tool in this study is the
spectral gap property for the operator $P$ when it holds. We use it
to obtain limit theorems, recurrence/transience property and
ergodicity for random walks on non compact extensions of the
corresponding dynamical systems.

DCDS

Let $( Ω , Α , \mathbb{P} , \tau )$ be an ergodic dynamical system.
The rotated ergodic sums of a function $f$ on $\Omega$ for $\theta
\in \mathbb{R}$ are $S_n^θ f : = \sum_{k=0}^{n-1} e^{2\pi i k \theta} f
\circ \tau^k, n \geq 1$. Using Carleson's theorem on Fourier
series, Peligrad and Wu proved in [14] that $(S_n^\theta
f)_{n \geq 1}$ satisfies the CLT for a.e. $\theta$ when $(f\circ
\tau^n)$ is a regular process.

Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by $K$-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to $\mathbb{Z}^d$-dynamical systems.

Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by $K$-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to $\mathbb{Z}^d$-dynamical systems.

DCDS

Let $(A_n(\omega))$ be a
stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Hölder function
$f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega)
\, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm \ mod \ }
1)$ 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 Komlòs and the
Multiplicative Ergodic Theorem.

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