Entropy Numbers of Some Ergodic Averages
Catherine Gamet, Michel Weber · Theory of Probability and Its Applications · 2000
In this paper we extend a recent remarkable covering numbers estimate for averages of contractions in a Hilbert space H due to Talagrand to some moving averages of contractions. By introducing a second regularization in Talagrand's spectral regularization, we find mild conditions on the spectral measure associated to any $x\in H$, allowing estimation of the number of Hilbertian balls of radius $0 < \varepsilon\le 1$, enough to cover the subset of H defined by $\{ B_n(x)=n^{-1} \sum_{j=n^2}^{n^2+n-1}U^{j}x$, $n\in \cal N\}$, where U is a contraction of H and $\cal N$ a geometric sequence. Moreover, we show that these conditions on the spectral measure ensure the existence of the modulus of continuity of $\{ T^{-1} \int_0^T f \circ U_t \, dt$, ${T\ge 1}\}$, where f is a contraction of $L^2(\mu )$ and $\{U_t,\ t\in \bf R \}$ is a flow which preserves the measure $\mu $. Finally, we give a covering numbers estimate in a non-Hilbertian case.