The CLT for rotated ergodic sums and related processes

Studia Mathematica, 2010

Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in L p , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in L p. For p = 1 we find that the maximal ergodic transforms are of weak type (1, 1). Convergence results are also proved. We give some general examples of Cesàro bounded semigroups. X f (τ t x)J t (x) dν(x) for all nonnegative measurable functions f and for all

Nonlinearity, 2013

We study the ergodic properties of compositions of interval exchange transformations and rotations. We show that for any interval exchange transformation T , there is a full measure set of α ∈ [0, 1) so that T • Rα is uniquely ergodic, where Rα is rotation by α.

Israel Journal of Mathematics, 2005

We prove extensions of Menchoff's inequality and the Menchoff-Rademacher theorem for sequences {f n } ⊂ L p , based on the size of the norms of sums of sub-blocks of the first n functions. The results are applied to the study of a.e. convergence of series n a n T n g n α when T is an L 2-contraction, g ∈ L 2 , and {a n } is an appropriate sequence. Given a sequence {f n } ⊂ L p (Ω, µ), 1 < p ≤ 2, of independent centered random variables, we study conditions for the existence of a set of x of µ-probability 1, such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random power series n f n (x)T n g converges π-a.e. The conditions are used to show that for {f n } centered i.i.d. with f 1 ∈ L log + L, there exists a set of x of full measure such that for every contraction T on L 2 (Y, π) and g ∈ L 2 (π), the random series n fn(x)T n g n converges π-a.e.

arXiv (Cornell University), 2017

We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than 1 2 are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property is well controlled almost everywhere. We prove that for any q-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.

Iternational Journal of Mathematics and Mathematical Sciences, 2006

We consider irrational rotations of the circle Tx = x + α (mod 1) and study the asymptotic behavior of sums of the type S n = n−1 k=0 π k with π k = ±1, the sign being determined by the location of T k x with respect to a binary partition.

In this paper we study the ergodic theory of a class of symbolic dynamical systems (Ω, T, µ) where T : Ω → Ω the left shift transformation on Ω = ∞ 0 {0, 1} and µ is a σ-finite T -invariant measure having the property that one can find a real number d so that µ(τ d ) = ∞ but µ(τ d−ǫ ) < ∞ for all ǫ > 0, where τ is the first passage time function in the reference state 1. In particular we shall consider invariant measures µ arising from a potential V which is uniformly continuous but not of summable variation. If d > 0 then µ can be normalized to give the unique non-atomic equilibrium probability measure of V for which we compute the (asymptotically) exact mixing rate, of order n −d . We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead d ≤ 0 then µ is an infinite measure with scaling rate of order n d . Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at z = 1 is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.

Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1980

Israel Journal of Mathematics, 2003

In this paper we continue our investigations of square function inequalities. The results in [9] are primarily one dimensional, and here we extend all the results to multi-dimensional averages. Our basic tool is still a comparison of the ergodic averages with various dyadic (reversed) martingales, but the Fourier transform arguments are replaced by more geometric almost orthogonality arguments. The results imply the pointwise ergodic theorem for the action of commuting measure preserving transformations, and give additional information such as control of the number of upcrossings of the ergodic averages. Related differentiation results are also discussed.

Journal of Fourier Analysis and Applications, 2008

We extend some results of C. Demeter about the boundedness of series of difference of convolutions to the setting of one-sided A p weights. We need to use a different approach. To be precise, we prove that the kernel of the associated operator satisfies a generalized Hörmander condition. The weighted inequalities allow us to transfer the result to the ergodic case, when the operator is induced by a mean bounded, invertible, positive groups.

1994