Subdiffusive behavior generated by irrational rotations
Résumé
We study asymptotic distributions of the sums $y_n(x) = \sum_{k=0}^{n-1} \psi(x+k\alpha)$ with respect to the Lebesgue measure, where $\alpha \in \mathbf{R} - \mathbf{Q}$ and where $\psi$ is the 1-periodic function of bounded variation such that $\psi (x) = 1$ if $x \in [0,1/2[$ and $\psi (x) = -1$ if $x\in[1/2,1[$. For every $\alpha\in\mathbf{R} - \mathbf{Q}$, we find a sequence $(n_j)_j \subset \mathbf{N}$ such that $y_{n_j}/\sqrt j$ is asymptotically normally distributed. For $n\ge 1$, let $z_n \in (y_m)_{m\le n}$ be such that $|| z_n ||_{\mathrm L^2} = \max_{m\le n} ||y_m||_{\mathrm L^2}$. If $\alpha$ is of constant type, we show that $z_n / || z_n ||_{\mathrm L^2}$ is also asymptotically normally distributed. We give an heuristic link with the theory of expanding maps of the interval.
Domaines
Probabilités [math.PR]
Origine : Fichiers produits par l'(les) auteur(s)
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