SPECTRAL REGULARIZATION IN ERGODIC-THEORY AND PROBABILITY

We show that the idea of spectral regularization introduced by Talagra nd in the study Of covering numbers of averages of contractions in a H ilbert space H can be concentrated in one inequality which turns out t o be a suitable tool for the study of other characteristics of the set of averages. This inequality generates an intrinsic Lipschitz embeddi ng of the circle and yields many useful corollaries. We also easily de duce the original Talagrand estimate of covering numbers and provide b etter estimates for geometric subsequences of the averages. Using majo rizing measuring technique, we prove a new criterion of the a.s. conve rgence of random sequences cinder suitable incremental conditions. We obtain as a corollary the classical theorem of Rademacher-Menshov on o rthogonal series and the famous spectral criterion. for the strong law of large numbers due to Gaposhkin.