A Markov operator P on a probability space (S, Σ, µ), with µ invariant, is called hyperbounded if... more A Markov operator P on a probability space (S, Σ, µ), with µ invariant, is called hyperbounded if for some 1 ≤ p < q ≤ ∞ it maps (continuously) L p into L q . We deduce from a recent result of Glück that a hyperbounded P is quasi-compact, hence uniformly ergodic, in all L r (S, µ), 1 < r < ∞. We prove, using a method similar to Foguel's, that a hyperbounded Markov operator has periodic behavior similar to that of Harris recurrent operators, and for the ergodic case obtain conditions for aperiodicity. Given a probability ν on the unit circle, we prove that if the convolution operator P ν f := ν * f is hyperbounded, then ν is atomless. We show that there is ν absolutely continuous such that P ν is not hyperbounded, and there is ν with all powers singular such that P ν is hyperbounded. As an application, we prove that if P ν is hyperbounded, then for any sequence (n k ) of distinct positive integers with bounded gaps, (n k x) is uniformly distributed mod 1 for ν almost every x (even when ν is singular). Definition. Let (S, Σ, µ) be a probability space and 1 ≤ p < ∞. A bounded operator T on L p (S, µ) is called hyperbounded if for some q > p the operator T maps L p (S, µ) into L q (S, µ). As observed in [28], a hyperbounded T maps L p to L q continuously, by the closed graph theorem. Note that if T maps L p to L ∞ , then it maps L p to L q for any p < q < ∞, since T f q ≤ T f ∞ ≤ C f p . Glück [28, Theorem 1.1] proved the following.
Let S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system ... more Let S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system of generators (A 1 , .. In particular, given a random walk on commuting matrices in SL(ρ, Z) or in M * (ρ, Z) acting on the torus T ρ , ρ ≥ 1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on T ρ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g. a torus or a connected extension of a torus), S a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on the spectral properties of the action of S, on random walks and on the variance of the associated ergodic sums.
Abstract. Let (Ω,A, P, τ) be an ergodic dynamical system. The rotated er-godic sums of a function... more Abstract. Let (Ω,A, P, τ) be an ergodic dynamical system. The rotated er-godic sums of a function f on Ω for θ ∈ R are Sθnf:= P n−1 k=0 e2piikθf◦τk, n ≥ 1. Using Carleson’s theorem on Fourier series, Peligrad and Wu proved in [14] that (Sθnf)n≥1 satisfies the CLT for a.e. θ when (f ◦ τ n) is a regular process. Our aim is to extend this result and give a simple proof based on the Fejèr-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 Zd-dynamical systems.
Let T and S be commuting contractions on a Banach space X. The elements of (I − T)(I − S)X are ca... more Let T and S be commuting contractions on a Banach space X. The elements of (I − T)(I − S)X are called double coboundaries, and the elements of (I − T)X ∩ (I − S)X are called joint cobundaries. For U and V the unitary operators induced on L 2 by commuting invertible measure preserving transformations which generate an aperiodic 2-action, we show that there are joint coboundaries in L 2 which are not double coboundaries. We prove that if α,β ∈ (0, 1) are irrational, with T α and T β induced on L 1 (Ì) by the corresponding rotations, then there are joint coboundaries in C(Ì) which are not measurable double cobundaries (hence not double coboundaries in L 1 (Ì)).
We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random fie... more We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a Z-random walk in different frameworks: probabilistic (when the r.f. is i.i.d. or a moving average of i.i.d. random variables) and algebraic (when the r.f. is generated by commuting automorphisms of a torus or by commuting hyperbolic flows on homogeneous spaces).
Let T and S be commuting contractions on a Banach space X . The elements of (I−T )(I−S)X are call... more Let T and S be commuting contractions on a Banach space X . The elements of (I−T )(I−S)X are called double coboundaries, and the elements of (I−T )X∩ (I − S)X are called joint cobundaries. For U and V the unitary operators induced on L2 by commuting invertible measure preserving transformations which generate an aperiodic Z-action, we show that there are joint coboundaries in L2 which are not double coboundaries. We prove that if α,β ∈ (0, 1) are irrational, with Tα and Tβ induced on L1(T) by the corresponding rotations, then there are joint coboundaries in C(T) which are not measurable double cobundaries (hence not double coboundaries in L1(T)).
We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random eld... more We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random eld (r.f.) along a 2d-random walk in dierent situations: when the r.f. is iid with a second order moment (random sceneries), or when it is generated by the action of commuting automorphisms of a torus. We consider also a quenched version of the FCLT when the random walk is replaced by a Lorentz process in the random scenery.
For N d-actions by algebraic endomorphisms on compact abelian groups, the existence of non-mixing... more For N d-actions by algebraic endomorphisms on compact abelian groups, the existence of non-mixing configurations is related to "S-unit type" equations and plays a role in limit theorems for such actions. We consider a family of endomorphisms on shift-invariant subgroups of F Z d p and show that there are few solutions of the corresponding equations. This implies the validity of the Central Limit Theorem for different methods of summation.
Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence o... more Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the power series in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also sup n n k=0 β k T k x < ∞ is equivalent to the convergence of the series. The last assertion is proved also when T is a mean ergodic contraction of L 1 . For normal operators on a Hilbert space we obtain a spectral characterization of the convergence of ∞ n=0 β n T n x, and a sufficient condition expressed in terms of norms of the ergodic averages, which in some cases is also necessary. For T Dunford-Schwartz of a σ-finite measure space or a positive contraction of L p , 1 < p < ∞, we prove that when {β k } is also completely monotone (i.e. a Hausdorff moment sequence) and For T a positive contraction of L p , p > 1 and f ∈ L p , we show that if the series ∞ n=0 (log(n+1)) β (n+1) 1-1/r T n f converges in L p -norm for some r ∈ ( p p-1 , ∞], β ∈ R, then it converges a.e.
Journal of Mathematical Analysis and Applications, 2020
Following Bermúdez et al. , we study the rate of growth of the norms of the powers of a linear op... more Following Bermúdez et al. , we study the rate of growth of the norms of the powers of a linear operator, under various resolvent conditions or Cesàro boundedness assumptions. We show that T is power-bounded if (and only if) both T and T * are absolutely Cesàro bounded. In Hilbert spaces, we prove that if T satisfies the Kreiss condition, ) for some ε > 0 (which depends on T ); if T is strongly Kreiss bounded, then T n = O((log n) κ ) for some κ > 0. We show that a Kreiss bounded operator on a reflexive space is Abel ergodic, and its Cesàro means of order α converge strongly when α > 1.
Let S be an abelian finitely generated semigroup of endomorphisms of a probability space (Ω, A, µ... more Let S be an abelian finitely generated semigroup of endomorphisms of a probability space (Ω, A, µ), with (T 1 , ..., T d ) a system of generators in S. Given an increasing sequence of domains After a preliminary spectral study when the action of S has a Lebesgue spectrum, we consider totally ergodic d-dimensional actions given by commuting endomorphisms on a compact abelian connected group G and we show a CLT, when f is regular on G. When G is the torus, a criterion of non-degeneracy of the variance is given.
Let S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system ... more Let S be an abelian group of automorphisms of a probability space (X, A, µ) with a finite system of generators (A 1 , ..., A d). Let A ℓ denote A ℓ1 1 ...A ℓ d d , for ℓ = (ℓ 1 , ..., ℓ d). If (Z k) is a random walk on Z d , one can study the asymptotic distribution of the sums n−1 k=0 f • A Z k (ω) and ℓ∈Z d P(Z n = ℓ) A ℓ f , for a function f on X. In particular, given a random walk on commuting matrices in SL(ρ, Z) or in M * (ρ, Z) acting on the torus T ρ , ρ ≥ 1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on T ρ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g. a torus or a connected extension of a torus), S a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on the spectral properties of the action of S, on random walks and on the variance of the associated ergodic sums.
Characteristic Functions, Scattering Functions and Transfer Functions, 2009
Let T be a normal contraction on a Hilbert space H. For f ∈ H we study the one-sided ergodic Hilb... more Let T be a normal contraction on a Hilbert space H. For f ∈ H we study the one-sided ergodic Hilbert transform lim n→∞ n k=1 T k f k . We prove that weak and strong convergence are equivalent, and show that the convergence is equivalent to convergence of the series . When H = (I -T )H, the transform is shown to be precisely minus the infinitesimal generator of the strongly continuous semi-group {(I -T ) r } r≥0 . The equivalence of weak and strong convergence of the transform is proved also for T an isometry or the dual of an isometry. For a general contraction T , we obtain that convergence of the series
Let (Ω, F , P) be a probability space, and let {Xn} be a sequence of integrable centered i.i.d. r... more Let (Ω, F , P) be a probability space, and let {Xn} be a sequence of integrable centered i.i.d. random variables. In this paper we consider what conditions should be imposed on a complex sequence {bn} with |bn| → ∞, in order to obtain a.s. convergence of P n Xn bn , whenever X 1 is in a certain class of integrability. In particular, our condition allows us to generalize the rate obtained by Marcinkiewicz and Zygmund when E[|X 1 | p ] < ∞ for some 1 < p < 2. When applied to weighted averages, our result strengthens the SLLN of Jamison, Orey, and Pruitt in the case X 1 is symmetric. An analogous question is studied for {Xn} an Lp-bounded martingale difference sequence. An extension of Azuma's SLLN for weighted averages of uniformly bounded martingale difference sequences is also presented. Applications are made also to modulated averages and to strong consistency of least squares estimators in a linear regression. The main tool for the general approach is (a generalization of) the counting function introduced by Jamison et al. for the SLLN for weighted averages.
Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence o... more Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hilbert transform limn P n k=1 T k x k . We prove that weak and strong convergence are equivalent, and in a reflexive space also < ∞ is equivalent to the convergence. We also show that - (which converges on (I -T )X) is precisely the infinitesimal generator of the semigroup (I -T ) r |(I-T )X .
In this paper we derive the spectral and ergodic properties of a special class of homogeneous ran... more In this paper we derive the spectral and ergodic properties of a special class of homogeneous random fields, which includes an important family of evanescent random fields. Based on a derivation of the resolution of the identity for the operators generating the homogeneous field, and using the properties of measurable transformations, the spectral representation of both the field and its covariance sequence are derived. A necessary and sufficient condition for the existence of such representation is introduced. Using an analysis approach that employs the solution to the linear Diophantine equations, further characterization and modeling of the spectral properties of evanescent fields are provided by considering their spectral pseudo-density function, defined in this paper. The geometric properties of the spectral pseudo-density of the evanescent field are investigated. Finally, necessary and sufficient conditions for mean and strong ergodicity of the first and second order moments of these fields are derived. The analysis, initially carried out for complex valued random fields, is later extended to include the case of real valued fields.
Discrete & Continuous Dynamical Systems - A, 2013
Let (Ω, A, P, τ ) be an ergodic dynamical system. The rotated ergodic sums of a function Using Ca... more Let (Ω, A, P, τ ) be an ergodic dynamical system. The rotated ergodic sums of a function Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that (S θ n f ) n≥1 satisfies the CLT for a.e. θ when (f • τ n ) is a regular process. Our aim is to extend this result and give a simple proof based on the Fejèr-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 Z d -dynamical systems.
Let T be a power-bounded operator on Lp(µ), 1 < p < ∞. We use a sublinear growth condition on the... more Let T be a power-bounded operator on Lp(µ), 1 < p < ∞. We use a sublinear growth condition on the norms { n k=1 T k f p} to obtain for f the pointwise ergodic theorem with rate, as well as a.e. convergence of the one-sided ergodic Hilbert transform. For µ finite and T a positive contraction, we give a sufficient condition for the a.e. convergence of the "rotated one-sided Hilbert transform"; the result holds also for p = 1 when T is ergodic with T 1 = 1. Our methods apply to norm-bounded sequences in Lp. Combining them with results of Marcus and Pisier, we show that if {gn} is independent with zero expectation and uniformly bounded, then almost surely any realization {bn} has the property that for every γ > 3/4, any contraction T on L 2 (µ) and f ∈ L 2 (µ), the series ∞ k=1 b k T k f (x)/k γ converges µ-almost everywhere. Furthermore, for every Dunford-Schwartz contraction of L 1 (µ) of a probability space and f ∈ Lp(µ), 1 < p < ∞, the series ∞ k=1 b k T k f (x)/k γ converges a.e. for γ ∈ (max{ 3 4 , p+1 2p }, 1].
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