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Randomly Perturbed Ergodic Averages

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Randomly Perturbed Ergodic Averages PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 8, Pages 224–244 (July 2, 2021) https://doi.org/10.1090/bproc/61 RANDOMLY PERTURBED ERGODIC AVERAGES JAEYONG CHOI AND KARIN REINHOLD-LARSSON (Communicated by Nimish Shah) Abstract. We consider a class of random ergodic averages, containing aver- ages along random non–integer sequences. For such averages, Cohen & Cuny 2 obtained uniform universal pointwise convergence for functions in L with max(1, log(1 + |t|))dμf < ∞ via a uniform estimation of trigonometric poly- 2 nomials. We extend this result to L functions satisfying the weaker condition max(1, log log(1+|t|))dμf < ∞. We also prove that uniform universal point- wise convergence in L2 holds for the corresponding smoothed random averages or for random averages whose kernels exhibit sufficient decay at infinity. 1. Introduction In this work, we adopt a general framework for describing random averaging operators. (Ω, F,P) denotes a probability space, and {νk}k∈Nr a sequence of inde- d d d pendent transition measures on Ω × R , νk :Ω× R → C . See Section 2 for a precise definition of independent transition measures. (X, D,m) denotes a proba- bility space and {Tt}t∈(R+)d a continuous semi–flow of positive isometries acting on 2 + d 2 L (X). Each measure νk(ω, .)on(R ) defines a bounded operator in L (X)asa weighted average over the flow: ω νk f(x)= Ttf(x)νk(ω, dt). Rd We denote their averages by ω 1 ω | Rd | (1.0.1) Kn f(x)= νk (f)(x), where bn = E( νk(., ) ). bn k∈[1,n]r k∈[1,n]r This approach includes averages of the form 1 1 1 T f(x), or T f(x)dt, r Xk(ω) r d Xk(ω)+t n n (2k(ω)) k∈[1,n]r k∈[1,n]r t <k(ω) + d + where Xk :Ω→ (R ) and k :Ω→ R are sequences of random variables; and averages of the form 1 Y (ω)T f(x), b k nk n ∈ r k [1,n] → C | | where Yk :Ω is a sequence of random variables with bn = k∈[1,n]r E( Yk ). Received by the editors October 10, 2018, and, in revised form, August 31, 2020, and September 11, 2020. 2020 Mathematics Subject Classification. Primary 37A05, 28D05. Key words and phrases. Random ergodic averages, universal convergence. c 2021 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0) 224 RANDOMLY PERTURBED ERGODIC AVERAGES 225 Before addressing the case of random averages, we review some key results for ergodic averages along subsequences. A sequence of bounded operators {Tn}n∈N on L2(X)hasthestrong sweeping out property if, for any >0, there exists a set E with 0 <m(E) <, such that lim supn→∞ Tn1E = 1 a.e. and lim infn→∞ Tn1E =0 a.e.. Let (X, D,m) be a non–atomic probability space and {τt}t∈R an aperiodic, er- godic, measure preserving flow on it. Ackoglu, Bellow, del Junco and Jones [1] { } { } showed that, for any increasing sequence of integers nk k∈N,if tk k∈N is a se- → 1 n quence such that tk 0, the averages Bnf(x)= n k=1 f(τnk+tk x)havethe strong sweeping out property and, therefore, there exist bounded functions for which pointwise convergence fails on sets of positive measure. Bergelson, Bosher- nitzan and Bourgain [7] had used Bourgain’s entropy method [11] to show that the ∞ averages Bnf diverge a.e. for some f ∈ L when the sequence {tk} is independent over the rationals. Ackoglu, del Junco and Lee [2] proved that the related averages, 1 n { } n k=1 f( x+tk ), for f in [0, 1), have the δ-sweeping out property. In [3], Ackoglu et al. showed that these averages have, in fact, the strong sweeping out property. When the derministic sequence {tk}k∈N is replaced by a random sequence, uniform positive results are obtained. Definition 1.1. A sequence of operators {Rn}n∈N of the form Rnf(x)= Ttf(x)μn(dt), Rd + d where {μn}n∈N is a sequence of finite measures on (R ) ,isuniversally good in 2 S ⊂ L , if for any probability space (X, D,m)andany{Tt}t∈(R+)d , a continuous 2 semi–flow of positive isometries acting on L (X), limn→∞ Rnf(x)existsa.e.for ∈ 1 any f S(X). When Rnf(x)= nr k∈[1,n]r Tnk f(x), we simply say the sequence + d {nk}k∈Nr ⊂ (R ) is universally good in S. Definition 1.2. With the notation from (1.0.1), we say the averaging operators 2 {Kn}n∈N are uniformly universally good in S ⊂ L if there exists Ω ⊂ Ω, with ∈ { ω} P (Ω ) = 1 such that, for all ω Ω , Kn k∈N is universally good in S.When → R+ d νk(ω, x)=δrk(ω)(x), with rk :Ω ( ) a sequence of random vectors; we simply say that the sequence {rk} is uniformly universally good in S. + In what follows, {nk}k∈N is a non–decreasing sequence in R , {γk}k∈N and {k}k∈N are independent sequences of random variables defined on a probability F { } ⊂ R+ space (Ω, ,P)and Tt t∈(R+) , a semi–flow of positive isometries. Reinhold [32] showed the averages 1 n 1 T f(x) dt, n k=1 2|k (ω)| |t|<|k (ω)| k+γk(ω)+t are uniformly universally good in Lp, p ≥ 1, provided e−1 ∈ Lq, 1 + 1 . Schneider k p q 1 n [38] showed that the averages Gnf(ω, x)= n k=1 Tnk+γk(ω)f(x), are uniformly 2 2 universally good in L for the sequence nk = k and γk i.i.d. random variables taking values 1 and −1 with probability 1/2. Schneider [39] extended this re- sult for universally good in L2 integer–valued sequences with the growth condition ks nk = O(2 ), for some s ∈ (0, 1), and where {γk}k∈N are integer–valued indepen- dent random variables. Duran and Schneider [22] applied the techniques to averages 1 n of the form n k=1 TXk(ω)f(x) where the distribution of the integer valued inde- pendent random variables {Xk}k∈N is generated by the convolution of a given law, ∗ P P (Sk) ≥ d Xk = d Y , for all k 1, (Y an integrable random variable). They also showed 226 JAEYONG CHOI AND KARIN REINHOLD-LARSSON 1 n 2 that the averages n k=1 Xk(ω)Tnk f(x) are uniformly universally good in L when ks nk = O(2 ) is a sequence of integers, without requiring it to be universally good 2 in L ,and{Xk}k∈N are centered i.i.d. with finite variance. Cohen and Cuny [15] obtained uniformly universally good results for the averages Gnf(ω, x) for sequences {γk}k∈N that do not take integer values. In light of [3, 2 7], pointwise convergence of Gnf(ω, x) fails for some functions in L but positive results are obtained by considering a subclass. Cohen and Cuny [15] proved, under certain condition on the sequences {nk}k∈N and {γk}k∈N, the averages {Gnf}k∈N 2 are uniformly universally good for f ∈ L such that log(2+|t|)dμf (t) < ∞.Their approach applies to a wider class of random averages as wells as to the study of convergence of certain random series. Definition 1.3. Let (Ω, F,P) be a probability space. We say the operators defined by the sequence of transition measures {νk}k∈Nr are uniformly norm summable in S ⊂ L2 if there exists a set Ω ⊂ Ω with P (Ω) = 1, such that, for every ω ∈ Ω, for any probability space (X, D,m), any continuous semi–flow of positive isometries 2 {T } ∈ R+ d in L (X), and any f ∈ S(X), t t ( ) νωf − E(ν )f k k ≤ C f , |k|r S k∈Nr 2 + d where |k| =max1≤i≤d |ki|.Whenrk :Ω→ (R ) is a sequence of random vari- { } ables and νk(ω, x)=δrk(ω)(x), we say that the sequence rk is uniformly norm– summable. Define (log ψ(t))+ =log ψ(t)ifψ(t) > 2, and 1 otherwise. 2 + 2 { ∈ 2 | | + ∞} Definition 1.4. (log ψ) L = f L : (log ψ t ) dμf < . For convenience, + 1/2 we’ll write f (log ψ)+L2 =( (log ψ|t|) dμf ) . + d Theorem 1.5 (Cohen and Cuny, [15, Theorem 4.12]). Let {Xn}n∈N ⊂ (R ) be α + d i.i.d random variables with E(|X1| ) < ∞ for some α>0.Let{nk}k∈N ⊂ (R ) , ∗ mβ with |nm| =maxk≤m |nk| = O(2 ),forsome0 <β<1.Then,{nk + Xk}k∈N is uniformly norm summable in (log)+L2. In particular, centered averages along the sequence {nk + Xk} 1 n (T f − E T f) n nk+Xk Ω nk+Xk k=1 are uniformly universally good in (log)+L2. In section 2, we present Cohen’s [14] extension of Cohen and Cuny’s [15] uniform estimates for almost periodic polynomials corresponding to the case of Fourier– Stieltjes transform of transition measures. Such estimates are essential for uniform universal results in L2. In Section 3, we prove the boundedness for a square function { ω − } { n ∈ N} for the centered averages Kn f EΩ(Knf) n∈I(ρ), where I(ρ)= ρ ,n (ρ>1); an estimate that suffices for a uniform universal result under the weaker + condition (log log(|t|)) dμf (t) < ∞.
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