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 averages 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 pointwise 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 probability 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

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

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 result 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 independent 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 independent 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) < ∞.

Deﬁnition 1.6. Given a set of complex numbers {xn}n∈I ,whereI is a countable index set, deﬁne its s–variation norm as ∞ { } | − |s 1/s xn n∈I v(s) =sup xnj xnj+1 j=1 RANDOMLY PERTURBED ERGODIC AVERAGES 227 where the supremum is taken over all possible non–decreasing subsequences {nj } in I. When the index set is N,wesimplywrite {xn}n∈N v(s) = xn v(s). Note that when the sequence is given by a sequence of Lebesgue measurable complex–valued functions {fn}n∈N,thenitss–variation norm ∞ | − |s 1/s fn(x) v(s) =sup fnj (x) fnj+1 (x) { { }} all non-decreasing nj j=1 is a Lebesgue measurable function. Variation norms are thus a useful tool for convergence, since the boundedness of the variation norm, fn(x) v(s) < ∞, implies convergence of {fn(x)}n.

Definition 1.7. A sequence of operators {Rn}n∈I as in Definition 1.1 is variationally good in S ⊂ L2, if, for some s>2, there exits a constant c(s) > 0such that, for any probability space (X, D,m)andany{Tt}t∈Rd a continuous semi–flow of positive isometries acting on L2(X) { } ≤ Rnf n∈I c(s) f S. v(s) 2 2 We say {Kn}n∈I is uniformly variationally good in S ⊂ L if there exists Ω ⊂ Ω, ∈ { ω} with P (Ω ) = 1 such that, for all ω Ω , Kn n∈I is variationally good in S. As before, when Rn or Kn are defined by a sequence (deterministic or random) we say that the sequence has the corresponding property. { } ⊂ R+ d Theorem 1.8. Let γnn∈N ( ) be independent random vectors such that, | | kβ ∞ { } ⊂ R+ d for some 0 <β<1, k≥1 P ( γk > 2 ) < .Letnk k∈N ( ) ,and β |n | = O(2k ). Then the sequence k 1 n (T f − E T f) n nk+γk Ω nk+γk k=1 n∈I(ρ) + 2 is uniformly universally good in (log log) L for any ρ>1. Moreover, when {γn} are i.i.d, 2 (a) if the sequence {nk}k∈N is universally good in L , then the sequence {nk + + 2 γk}k∈N is uniformly universally good in (log log) L ;and 2 (b) if in addition the sequence {nk}k∈I(ρ) is variationally good in L for ρ>1, then the the sequence {nk + γk}k∈I(ρ) is uniformly variationally good in (log log)+L2.

In Section 3 we prove a general version of this theorem for averages {Kn} of transition measures. There is a large literature on sequences with various growth conditions which are universally good in L2 and thus, this theorem gives conditions under which such sequences can be randomly perturbed with real valued sequences and still main- tain uniform convergence on a subclass of L2. Full variational inequalities along subsequences have been more diﬃcult to obtain. The technique of handling variations to prove pointwise convergence was introduced in ergodic theory by Bourgain [10] for the regular ergodic averages. The methods were applied to sequences of primes and polynomial sequences [10]. Recently, Krause [27] showed the variational inequality for polynomial sequences and Zorin–Kranich [43] showed variational inequalities for weighted averages of powers of primes and for the polynomial sequence 228 JAEYONG CHOI AND KARIN REINHOLD-LARSSON

(n, n2,...,nd) ∈ Zd. Therefore Theorem 1.8 yields uniform universal convergence results for the corresponding randomly perturbed averages. { } ⊂ R+ Corollary 1.9. Let γn n∈N be i.i.d. random variables such that, for some | | kβ ∞ 0 <β<1, k≥1 P ( γk > 2 ) < .Letp(x) be a polynomial with integer coeﬃ- cients. Then the sequences {p(n)+γn}n∈N and {γn}n∈N, are uniformly universally good in (log log)+L2. { } In particular, if the γkk∈N take value in [0, 1], the corresponding random Bel- 1 n low’s problem averages n k=1 Tγk(ω)f(x), are uniformly universally and variationally good in (log log)+L2. From Krause [27] and Zorin–Kranich [43] the next two applications follow.

+ Corollary 1.10. Let λ(x) be the von Mangoldt function and let {γn}n∈N ⊂ R β be i.i.d. random variables such that, for some 0 <β<1, P (|γ | > 2k ) < k≥1 k ∞ 1 N . Then the averages N k=1 λ(k)Tk+γk(ω)f(x) are uniformly universally and variationally good in (log log)+L2. { } ⊂ R+ d Corollary 1.11. Let γn n∈Nd ( ) be i.i.d. random vectors such that, for | | kβ ∞ { 2 d some 0 <β<1, k≥1 P ( γk > 2 ) < . Then the sequence (n, n ,...,n )+ + 2 γn(ω)} is uniformly universally and variationally good in (log log) L . In Section 4, full L2 result are obtained when the kernels associated with the averaging operators have additional smoothing properties. Rd → R Let ζ : be positive, integrable with Rd ζ(t)dt = 1, and with support | |≤ 1 t1 td on t 1. Let Lf(x)= d ζ(t)Ttf(x)dt where ζ(t)= ζ( ,..., ). Let R 1···d 1 d d d {γk}k∈Nr ⊂ R and {k}k∈Nr ⊂ (0, 1] be independent sequences of independent + d positive random vectors, and {nk}k∈Nr ⊂ (R ) .Fromhereon,Fnf(x) denotes the following smoothed average around the observations nk + γk(ω), 1 (1.11.1) F f(ω, x)=F f(x)= T L f(x). n n nr nk+γk(ω) k(ω) k∈[1,n]r

The regular averages along the sequence {nk}k∈Nr are denoted as 1 (1.11.2) A f(x)= T f(x). n nr nk k∈[1,n]r

d d Theorem 1.12. Let {γk}k∈Nr ⊂ R and {k}k∈Nr ⊂ (0, 1] be two independent + d sequences of positive, independent random vectors, and {nk}k∈Nr ⊂ (R ) . Assume they satisfy the following conditions: | |−α ∞ ∈ Nr (i) E(min1≤j≤d k,j )< for some α>0 (and any k ); r−1 | | jrβ ∞ (ii) for some 0 <β<1, j≥1 j P ( γj > 2 ) < ; | | |k|rβ (iii) nk =O(2 );and d | |α | | ∞ (iv) supt j=1 max(1, tj ) ζ(t) < .

{ ∗ } r Then the sequence of transition probability measures δnk+γk ζk k∈N is uniformly 2 norm summable in L . Moreover, when {k} and {γk} are i.i.d., if the sequence 2 {nk} is universally good in L , then the averaging operators {Fn}n∈N are uniformly 2 2 universally good in L ; if in addition {Anf} are variationally good in L ,then 2 {Fn}n∈N is uniformly variationally good in L . RANDOMLY PERTURBED ERGODIC AVERAGES 229

This theorem also holds for more general transition measures {νn} and their averages {Kn}, under equivalent requirements. See Theorem 4.2.

Example 1.13. Let d, r =1.Let{k}k∈N ⊂ N be positive i.i.d with E(1/1) < ∞, { } ⊂ R | | | |≤ and let γk k∈N be bounded i.i.d. Let ζ = χ[0,1], then supt max(1, t ) ζ(t) 1. From Krause [27], if p(x) is a polynomial with integer coefficients, then the averages { ∗ } Fn defined by the sequence of transition probability measures δp(k)+γk ζk k∈N { ∗ } or the smoothed–Bellow problem averages defined by δγk ζk k∈N are uniformly universally good in L2. d Similarly, with d ≥ 1, {k}k∈N ⊂ N positive i.i.d with E(min1≤j≤d 1/k,j) < ∞, d {γk}k∈N ⊂ R bounded i.i.d., and let ζ = χ[0,1]d .Wehave sup max(1, |tj |)|ζ(t)|≤1. t 1≤j≤d

Thus, from Zorin–Kranich [43], the averages Fn deﬁned by the sequence of transition { 2 d ∗ } ∈N probability measures δ(k,k ,...,k )+γk(ω) ζk(ω) k are uniformly universally and variationally good in L2.

2. Uniform estimates Estimates for random trigonometric polynomials have been essential in proving convergence of random Fourier series as well as ergodic averages along subsequences and modulated ergodic averages. Paley and Zygmund(1930–32) [30] and Salem and Zygmund (1954) [37] provided the first estimates for trigonometric sums in ∞ ikx { } their study of Fourier series with random signs: k=1 kcke where the k is a Rademacher sequence, and {ck} is a sequence of complex numbers. They were also used to prove convergence of random Fourier and almost periodic series [14–16, 19, 23, 41]. In ergodic theory, their study yielded applications to the convergence of averages along subsequences and averages with random weights. Bourgain [9–12] used them to prove pointwise convergence of ergodic averages along polynomial sequences and Bourgain and Wierdl [42] applied them to pointwise convergence of ergodic averages along sequences of primes. Bourgain, Bergelson and Boshenitzan [7] use them to prove pointwise convergence of ergodic averages with random weights as well as Assani [4, 5], Rosenblatt and Wierdl [35], and Cohen and Lin [18]. Schneider [39] used them to prove convergence of ergodic averages along perturbed sequences of squares, with integer perturbations. Estimates on the associated trigonometric polynomials was obtained by means of an estimate on the derivative of those polynomials as in Kahane [26]. Cohen and Cuny [16] extended the estimates of Salem and Zygmund to obtain uniform esti- n i t,αk mates of multidimensional random exponential sums of the form k=1 Xk e , d where {Xn} is a sequence of random variables, {αk}⊂R are sequences of real numbers, and t ∈ Rd. Cohen [14] then adapted it to weighted sums of finite transition measures as stated in Theorem 2.3. Recent works have also used such estimates to study convergence of power series of isometries with random coefficients, including Assani [6], Boukhari and Weber [8], Cohen and Lin [18], Cohen and Cuny [15, 16] and Cohen [14]. d Notation. Henceforth, we denote the inner product for vectors u, v in R as u, s = d | | i=1 uisi; the coordinate–wise product as u.s =(u1s1,...,udsd), and u = max1≤i≤d |ui|. 230 JAEYONG CHOI AND KARIN REINHOLD-LARSSON

Definition 2.1. Let (Ω, F,P) be a probability space, and B the Borel sigma– algebra on Rd, d ≥ 1. A function ν :Ω× Rd → Cd is a finite complex valued transition measure on Ω × Rd if (i) ν(., B)isanF–measurable function for any B ∈B; (ii) ν(ω, .) is a finite complex valued measure on B, for any ω ∈ Ω; d (iii) and EΩ(|ν(.)|) < ∞,where|ν(ω)| := |ν|(ω, R ) denotes the variation norm of the measure ν(ω, .)andEΩν denotes the measure, on B, defined by ∈B EΩν(B)= Ω ν(ω, B)dP , for any B .

Notation. We will say that a sequence of transition measures {νk} is bounded if | | ∞ ∞ supk νk L (Ω) < .

Definition 2.2. The sequence {νk}k∈I of transition measures is independent if d for every finite set of Borel measurable simple functions g1,...,gm on R ,and any finite set k1,k2,...,km of pairwise distinct indices in I, the random variables { } Rd gi(x)νki (ω, dx) i=1,...,m are independent. Given a transition measures ν, its Fourier–Stieljes transform is ω i t,u νk (t)= e ν(ω, du). Rd Uniform estimates that control linear sums of transition measures were considered by Cohen in [14]. { } Theorem 2.3 (Theorem 2.8 in [14]). Let Lk k∈N be a sequence of positive num- ≥ ∞ ∞ 2 ∞ m 2 bers, Lk 1, such that n=1 m=n+1(1/Ln,m) < ,withLn,m = k=n+1 Lk. Let {νk}k∈N be a sequence of independent, finite complex valued transition measure on Ω ×B with |νk| L∞(Ω) < ∞, for all k ≥ 1.Let m i t,u i t,u Pn,m(t)= e νk(du) − e EΩνk(du) − d − d k=n+1 [ Lk,Lk] [ Lk,Lk] be the sum of the difference of the (truncated) Fourier–Stieljes transform corre- { } sponding to the measures νk(., t) and their expected values. And let Vn,m = m | | 2 { } k=n+1 νk L∞(Ω).Then,thereexists>0 and C>0, independent of νk , such that 2 max ∈ − d |Pn,m(ω, t)| sup sup exp( t [ T,T] ) n T ≥2 d+2 Vn,m log(Ln,m T ) L1(Ω) An immediate consequence is the following proposition. The notation ϕ(x) |x| means there exists a constant c>0 such that ϕ(x) ≥ c|x|.WeuseC(.)orc(.) to denote functions that depend on the indicated parameter(s), whose values may change in the different instances. Proposition 2.4. Let ϕ : R → R+ be non–decreasing with ϕ(x) |x|.Let {νk}k∈Nr be independent complex valued transition measures on Ω×B,and{ak}k∈Nr a sequence in (0, 1]. Assume that akEΩ|νk|(|t| >ϕ(|k|)) < ∞. k∈Nr RANDOMLY PERTURBED ERGODIC AVERAGES 231

r + Let Nn,m = {k ∈ N : n<|k|≤m}. Then there exists C :Ω→ R ﬁnite P–a.e. such that, for all m, 2 ω − maxt∈[−T,T]d k∈N ak(νk (t) EΩνk(t)) (2.4.1) sup sup n,m ≤ C(w). 2 2 m>n T>2 1+ a |ν | ∞ log(max(ϕ(m),T)) k∈Nn,m k k L (Ω)

In particular, if {νk}k∈Nr are bounded then 2 ω max ∈ − d a (ν (t) − E ν (t)) t [ T,T] k∈Nn,m k k Ω k sup sup ≤ C(w). m>n T>2 a2 log(max(ϕ(m),T)) k∈Nn,m k

Proof. Theorem 2.3 can be applied to the transition measures wk = akνk,now r indexed over k ∈ N , by simple re–numbering of the index set, with Lk = ϕ(|k|). We have 2 ∼ r−1 2 r 2 r+2 Ln,m = Lk j ϕ (j) m ϕ(m) ϕ(m) n<|k|≤m n

− d ω − Let Jn =[ ϕ(n),ϕ(n)] and Dνk(ω, t)=ak(ν k(t) EΩνk(t)). Decompose Dν (ω, t)=P + Q , where k∈Nn,m k n,m n,m i t,u i t,u Pn,m(ω, t)= ak e νk(ω, du) − e EΩνk(du) , J|k| J|k| k∈Nn,m and i t,u i t,u Qn,m(ω, t)= ak e νk(ω, du) − e EΩνk(du) . Jc Jc k∈Nn,m |k| |k|

1 By assumption on {νk},thereexistsC2(ω) ∈ L (Ω) such that

sup sup |Qn,m(ω, t)| C2(ω). m>n≥1 t Combining both estimates, (2.4.1) is obtained. 232 JAEYONG CHOI AND KARIN REINHOLD-LARSSON

+ d Example 2.5. Let {nk}k∈Nr ⊂ (R ) ; {γk}k∈Nr be independent bounded random ∞ ∞ { } vectors such that supk γk L = B< ; and let Xk k∈Nr be i.i.d complex valued { } r random variables, independent of γk k∈N , and let νk(ω, A)=Xk(ω)δnk+γk(ω)(A), A ∈D.Now ω − Pm(ω, t)= (νk (t) EΩνk(t)) k∈[1,m]r i t,nk i t,γk(ω) i t,γk = e Xk(ω)e − EΩ(X1)EΩ(e ) . k∈[1,m]r

| | |k|rβ ∞ |x|rβ If supk nk /2 = c< ,forsome0<β<1, using ϕ(x)=(1+c)2 ,the assumption of Proposition 2.4 is satisﬁed by a tail distribution condition on the γk’s | ω| | | |k|rβ ≤ | | | | |k|rβ ∞ EΩ νk ( t > (1 + c)2 ) EΩ( X1 ) P ( γk(ω) > 2 ) < . k∈Nr k∈Nr Then, by Proposition 2.4, there exists C(ω) ∈ L1(Ω), such that |P (ω, t)|2 sup sup max m = C(ω) < ∞. r mrβ m≥1 T ≥2 t∈[−T,T]d 1+m log(max(T,2 ))

1 ≤ nrβ In particular, for Dn(ω, t)= nr Pn(ω, t)andT 2 ,weobtain

nrβ | |2 log(max(T,2 )) ≤ 1 max Dn(ω, t) C(w) C(w) − . t∈[−T,T]d nr nr(1 β)

3. Applications to ergodic theory { } In this section, νk k∈Nr denotes a sequence of independent transition measures ×B | | 2 | | Rd ∞ on Ω , Vm = k∈[1,m]r νk L∞(Ω) and bm = k∈[1,m]r νk (., ) L (Ω).The averages ω 1 ω Kn f(x)=Knf(x)= νk (f)(x) bn k∈[1,n]r have associated kernels ω 1 1 i t,u Kn (t)=Kn(t)= νk(ω, t)= e νk(ω, du). bn bn k∈[1,n]r k∈[1,n]r Next, consider the averages deﬁned by their expected values 1 Enf(x)=EΩ(Knf(x)) = EΩνkf(x) bn k∈[1,n]r and its associated kernel 1 ω 1 i t,u En(t)=EΩ(Kn(t)) = EΩ(νk )(t)= e EΩνk(du). bn bn k∈[1,n]r k∈[1,n]r Let Dnf(x)=Knf(x)−Enf(x)andDn(t)=Kn(t)−En(t) denote the corresponding centered averages and their kernels. RANDOMLY PERTURBED ERGODIC AVERAGES 233

Note 3.1. A family {Tt}t∈(R+)d is a continuous parameter semi–ﬂow of isometries + d acting on a Hilbert space H or an (R ) –action, if for all x ∈ H, T0x = x , + d TsTtx = Ts+tx for s, t ∈ (R ) ,themapt → Ttx, x is continuous, and Tt is an isometry for each t ≥ 0. A classical generalization of Stone’s Theorem [34] provides a spectral repre- d d sentation for unitary actions of R .Thatis,if{Tt}t∈Rd is a R –action by unitary operators on a Hilbert space H, then for any x ∈ H, there is a positive d d ﬁnite measure μx on R , called its spectral measure, such that for any s ∈ R , i s,t Tsx, x = Rd e dμx(t). Such result extends to Rd isometry actions via the following unitary dilation theorem (Principal Theorem in Appendix to Riesz–Nagy Functional Analysis [34]). d Theorem 3.2 (Riesz–Nagy, [34]). Let {Tt}t∈Rd be a representation of R by isome- d tries on a Hilbert space H. There exist a Hilbert space H2 ⊃ H and an R action {Ut}t∈Rd by unitary operators on H2,suchthat,ifπ : H2 → H is the orthogonal d projection, then πUtx = Ttx, for all t ∈ R and x ∈ H.

Remark. Norm estimates can be extended to actions by contractions in the case d = 1, via Sz. –Nagy unitary dilation theorem (Szökefalvi–Nagy and Foias [40]). This still may hold for d = 2 but not in higher dimensions. Parrott [31] gave an example where the dilation theorem is no longer true in the case of N3 actions by contractions. Remark. For actions by semigroup of isometries, the spectral representation allows us to estimate norms for operators defined by a finite measure on the semigroup. With the notation of Riesz–Nagy theorem and the spectral measure, if ν a finite measure on (R+)d, by standard arguments we conclude 2 2 2 Ttx ν(dt) ≤ Utx ν(dt) = |ν(t)| dμx(t). Rd H Rd H Rd

Notation. For any positive real number v, abusing notation we write Kvf = K v f, Dvf = D v f, bv = b v , Vv = V v , etc.. Also, ϕ is a continuous strictly increasing positive function on R,andψ its inverse ψ(y) = inf{x ≥ 0:ϕ(x) >y}.

Theorem 3.3. Let {νk}k∈Nr be a sequence of independent, ﬁnite complex valued transition measures on Ω ×B.Letϕ be a non–decreasing positive function such that ϕ(|x|) |x|,andψ denotes its inverse. Assume that 1 1 (a) sup |νk(ω)|∈L (Ω), n bn k∈[1,n]r (b) EΩ|νk|(|t| >ϕ(|k|)) < ∞,and k∈Nr n 1+Vρn log(ϕ(ρ )) ∞ (c) c(ρ)= 2 < , for all ρ>1. b n n≥1 ρ Then, there exists 0 1 and any probability space (X, D,m) with any continuous semi–ﬂow of isometries {Tt}t∈(R+)d acting on it, if f ∈ L2(X), ∞ | |2 Dρn f

| |2 2 max|t|≤ϕ(ρn) Dρn (t) bρn C1(ω)= sup n ρ>1,n 1+Vρn log(ϕ(ρ )) is ﬁnite P–a.e. Therefore, by assumption (c), ∞ n 1+Vρn log(ϕ(ρ )) 2 2 I C1(ω) 2 f 2 = C1(ω)c(ρ) f 2. b n n=1 ρ For the second term II, by assumption (a) 1 1 sup sup |Dn(t)|≤sup |νk(ω)| = C2(ω) ∈ L (Ω). n t n bn k∈[1,n]r Thus ∞ ∞ | |2 ≤ 2 II = Dρn (t) dμf C2 (ω) k dμf k | |≤ (k+1) k | |≤ (k+1) n=1 k≥n ϕ(ρ )< t ϕ(ρ ) k=1 ϕ(ρ )< t ϕ(ρ ) ∞ 2 2 ≤ C2 (ω) | | C2 (ω) +| | log2 ψ( t )dμf (log ψ) t dμf . log ρ k | |≤ (k+1) log ρ 2 k=1 ϕ(ρ )< t ϕ(ρ ) 2 Combining the estimates for both terms I and II, the proposition is proven.

We are interested in the cases where {νk} are transition probability measures, thus condition (a) in this theorem is immediately satisﬁed. Condition (c) is fulﬁlled by the choice of growth function ϕ. Thus, the restriction on the estimate for the square function on {Dρn f} comes from the distribution of the transition probability measures {νk}.

Proposition 3.4. Let {νk}k∈Nr be a sequence of independent, bounded transition measures on Ω ×B such that, for some 0 <β<1 and some constant c>0, |k|rβ EΩ|νk|(|t| >c2 ) < ∞. k∈Nr Then there exists 0 1. n=1 2 RANDOMLY PERTURBED ERGODIC AVERAGES 235

rβ Proof. Choosing ϕ(x)=c 2x ,forx ≥ 0, it suﬃces to check the assumptions of { } | | Theorem 3.3. Since νk k∈Nr are bounded transition measures, supω∈Ω supk νk(ω) r r = K<∞.Thus,bn ∼ n and Vn ∼ n and 1 sup sup |νk(ω)|∼K ω∈Ω n bn k∈[1,n]r satisfying assumption (a). Assumption (b) becomes the assumption of this proposition, and assumption (c) is satisﬁed for β ∈ (0, 1)

n rn ρrβn 1+Vρn log(ϕ(ρ )) ∼ ρ log(2 ) c(ρ, β)= 2 2rn b n ρ n≥1 ρ n≥1 1 = < ∞ for all ρ>1. ρ(1−β)rn n≥1

After the square function result along exponential subsequences I(ρ)={ρn} is obtained for all ρ>1, Theorem 3.3 or Proposition 3.4, one is tempted to prove a variational inequality for averages with kernels Kn. But without additional control on the decay of the kernels, this approach may fail to be fruitful. However, this result is enough for universal pointwise convergence of the averages. The steps for proving convergence were inspired by [35] and related works. Proposition 3.5 (Properties of variation norms). (a) For s ≥ 1, . is a semi-norm; v(s) ≤ ∞ | |s 1/s (b) xn v(s) 2 n=1 xn ;and ≤ { ≤ } (c) xn v(s) 2 k xn : nk n

xnk =0for all k. A proof of these properties can be found in [17] along with some discussion of the applications of the variation norm to ergodic theory, see also [24, 25]. A complex valued transition measure ν is dominated by a positive transition measure, deﬁned by its variation norm, which by abusing notation we denote by |ν|, |ν|(ω, B)=|ν(ω, B)| = variation norm of ν(ω, .) restricted to the set B,B ∈B. Continuing abusing notation, we denote | ω| 1 | |ω | | 1 | | Kn f(x)= vk f(x)and En f(x)= EΩ vk f(x). bn bn k∈[1,n]r k∈[1,n]r Theorem 3.6. With the assumptions and notation of Theorem 3.3. + 2 (a) {Kn − EΩKn}n∈I(ρ) are uniformly universally good in (log ψ) L for any ρ>1. 2 (b) If for some ρ>1, the averages {En}n∈I(ρ) are universally good in L ,then + 2 the averages {Kn}n∈I(ρ) are uniformly universally good in (log ψ) L . 2 (c) If for some ρ>1, {En}n∈I(ρ) are variationally good in L ,then{Kn}n∈I(ρ) are uniformly variationally good in (log ψ)+L2. 236 JAEYONG CHOI AND KARIN REINHOLD-LARSSON

2 (d) Assume {En}n∈I(ρ) are universally good in L for all ρ>1,and

lim sup bρn+1 /bρn =1. ρ→1 n

If {νk}k∈Nr are real–valued positive transition measures or {|En|}n∈I(ρ) are 2 also universally good in L ,then{Kn}n∈N is uniformly universally good in (log ψ)+L2. Proof. By Theorem 3.3, there exists 0 1. 2 Part (c): Since {En}n∈I(ρ) are variationally good in L ,thereexistss>2and c (s, ρ) > 0 such that Eρn f v(s) 2 ≤ c (s, ρ) f 2.Forω ∈ Ω , by Proposition 3.5,

Kρn f v(s) 2 ≤ Eρn f v(s) 2 + Dρn f v(s) 2 2 ≤ c (s, ρ) f 2 +2 |Dρn f| 2 n 1/2 + ≤ (c (s, ρ)+2C(ω)c(ρ)) (log ψ) |t|dμf .

Part (d):

Case 1({νk}k∈Nr are real–valued positive transition measures). In this case, since Tt’s are positive operators, Knf are themselves positive operators. Consider the 1/i sequence of ρi =2 decreasing to 1. Since {En}n∈I(ρ) are universally good in 2 L for all ρ>1, and by (b), for any ω ∈ Ω and any i, K n f converges almost ρi everywhere for any f ∈ (log ψ)+L2. It suﬃces to consider 0 ≤ f ∈ (log ψ)+L2, ω ω a.e.. Notice that with our choice of ρi, limn→∞ K n f(x) = limn→∞ K n f(x)= 2 ρi limn→∞ E2n f(x) a.e., for any i ≥ 1, and so we denote this limit as Lf(x). Note b n+1 ρ i n n+1 that this limit does not depend on ω. Letting Mi =sup ;forρ ≤ u<ρ , n bρn i i i we have

1 bρn bρn+1 ω ≤ i ω ≤ ω ≤ i ω ≤ ω Kρn f(x) Kρn f(x) Ku f(x) Kρn+1 f(x) MiKρn+1 f(x). Mi i b n+1 i bρn i i ρi i Then 1 Lf(x) ≤ lim inf Kωf(x) ≤ lim sup Kωf(x) ≤ M Lf(x), →∞ u u i Mi u u→∞ and since Mi −−−→ 1, i→∞ lim Kuf(x)=Lf(x) a.e.. u→∞ A similar relationship can be established for the averages Enf,thusLf(x)= ω limn→∞ Kn f(x) = limn→∞ Enf(x) a.e.. RANDOMLY PERTURBED ERGODIC AVERAGES 237

Case 2 (General transition measures). The conditions of this theorem on the transition measures {νk}k∈Nr are actually conditions on {|νk|}k∈Nr . Therefore, if both 2 {Eρn } and {|Eρn |} are universally good in L for any ρ>1, the results (a), (b) {| ω| } {| |} { } and (d) part (i) apply to the averages Kn f and thus both Kn and Kρn are uniformly universally good in (log ψ)+L2 (for any ρ>1). 1/i As before, consider the sequence of ρi =2 , decreasing to 1. Let Ω ⊂ Ωof ω ω probability 1, such that, for any ω ∈ Ω , {|K |} and {K n } are universally good n ρi in (log ψ)+L2 (for all i). Again, it suﬃces to consider 0 ≤ f ∈ (log ψ)+L2,andwe ω ω have limn→∞ E2n f(x) = limn→∞ K n f(x) = limn→∞ K n f(x)=Lf(x) a.e., for 2 ρi ≥ n ≤ n+1 any i 1. For ρi u< ρi , bu − bρn 1 | ω − ω |≤ i | ω | | |ω Ku f(x) Kρn f(x) Kρn f(x) + νl f(x) i bu i bu l: ρn <|l|≤u i − bρn+1 bρn bρn+1 ≤ i i | ω | i | ω | −| ω | Kρn f(x) + Kρn+1 f(x) Kρn f(x) bρn i bρn i i i i bρn+1 ≤ i − | ω | | ω | 1 Kρn f(x)+ Kρn+1 f(x) bρn i i i | ω | −| ω | + K n+1 f(x) Kρn f(x) . ρi i

b n+1 ρ i ω + 2 Since limi→∞ sup =1and{|K |} is universally good in (log ψ) L , n bρn n i ω ω ω limi→∞ lim sup →∞ |K f(x) − K n f(x)| = 0 a.e.. thus limu→∞ K f(x)=Lf(x) u u ρi u + 2 a.e., establishing that {Kn} are uniformly universally good in (log ψ) L .

Remark. Note that if the transition measures {νk} are complex valued with EΩνk = + 2 0, then, uniformly, Knf → 0 a.s. as n −−−−→ ∞,forf ∈ (log ψ) L . n∈I(ρ) Remark. The result of this theorem can be improved to hold in L2 with an additional requirement on the Dn kernels. See Section 4. Proof of Theorem 1.8. It follows as a corollary of Proposition 3.4 and 3.6, noticing { } ∈ Nr that, when γk are i.i.d., then Enf = An(EΩTγe f)(foranye ). Example 3.7 (Essentially the proof of Corollary 1.9). Consider again Example 2.5 with {nk}k∈N an integer valued polynomial sequence (nk = p(k) for some integer– 2 coeﬃcient polynomial p(x)), or any universally good sequence in L .Let{γk}k∈N, {Xk}k∈N be independent sequences of i.i.d. real–valued variables with EΩ(|γ1|) < ∞ | | ∞ and EΩ( X1 ) < .Letνk(ω, .)=Xk(ω)δnk+γk(ω).Then |k|β kβ EΩ|νk|(|t| >c2 )= EΩ(|Xk|)P (|γk| >c2 ) k∈N k∈N 1/β

EΩνkf = EΩ(Xk)Tnk EΩTγk(ω)f = Tnk g, 238 JAEYONG CHOI AND KARIN REINHOLD-LARSSON

∈ 2 ∈ 2 | | where g = EΩ(Xk)EΩTγ1 f L (X)iff L (X). Thus Enf = Ang, En f = Anh, | | | | 2 h = EΩ(Xk) EΩTγ1 f, and therefore, En and En are universally good in L . 1 n Thus, by Theorem 3.6, the averages n k=1 Xk(ω)Tnk+γk(ω)f(x) are uniformly universally good in (log log)+L2. Also, from the proof,

1 n lim Xk(ω)Tn +γ (ω)f(x)=EΩ(X1) lim An(EΩTγ f)(x). n→∞ n k k n→∞ 1 k=1

4. Smoother averages ω We now consider the particular example of averages Fn f(x) deﬁned in (1.11.1). Their corresponding kernels ω 1 i t,(n +γ (ω)) F (t)= ζ(t. (ω))e k k n nr k k∈[1,n]r

ω ω decompose as Fn (t)=En(t)+Dn (t)whereEn(t)=EΩFn(t)and ω 1 i t,γ (ω) i t,γ i t,n D (t)= e k ζ(t. (ω)) − E (e k )E (ζ(t. )) e k . n nr k Ω Ω k k∈[1,n]r

+ d d Theorem 4.1. Let {γk}k∈Nr ⊂ (R ) and {k}k∈Nr ⊂ (0, 1] be two independent +d sequences of positive i.i.d. random vectors, and {nk}k∈Nr ⊂ R . Assume that | | −α ∞ ∈ Nr (a) EΩ ((min1≤j≤d e,j ) ) < , for some α>0 and some e ; | | |k|rβ ∞ ≥ (b) k∈Nr P ( γk >c2 ) < for some constants c 1,and0 <β<1; and |k|rβ d (c) |nk| = O(2 ). Let ζ : R → R be positive, integrable with unit integral, with support on {|t|≤1}, and satisfying d α (d) sup max(1, |tj |) |ζ(t)| < ∞. t j=1 2 If {An}n∈N is variationally good in L then {Fn}n∈N is uniformly variationally good in L2.

r Proof. Fix e ∈ N .Sinceboth{k} and {γk} are i.i.d. and independent of each other, ω Enf = EΩ(Fn f)= EΩ(ζe (t))TtAn(EΩ(Tγe f))dt. (R+)d

2 Since {An}n∈N is variationally good in L , there exists a constant c(s) > 0such that ≤ ≤ Enf v(s) EΩ(ζe (t)) An(Tγe+tf) v(s) dt c(s) f 2. 2 (R+)d 2 { ω ω − } Next, we consider the centered averages Dn f = Fn f Enf along a well chosen ≥ { } subsequence. For h 1, let aj j∈Ih to be the sequence of equally spaced integers h h+1 h 2 such that 2 ≤ aj < 2 , aj+1 − aj ∼ 2 /h , and the smallest element in this RANDOMLY PERTURBED ERGODIC AVERAGES 239

h sequence is 2 . Rename this sequence {ml} = ∪k ∪j∈I aj . h 2 2 ω ≤ ω ≤ ω 2 Dm f v(s) Dm f v(2) 4 Dm f 2 l 2 l 2 l l≥1 ≤ | ω |2 4 Daj (t) dμf |t|≤22rβh h≥1 j∈Ih | ω |2 +4 Daj (t) dμf =I+II. |t|>22rβh h≥1 j∈Ih To estimate the ﬁrst term I we need ﬁrst to check the requirements of Proposition ⊂ Rd 2.4. For A (Borel measurable) let ζk(ω)(A)= A ζk(ω)(t)dt.Notethat since the support of ζ ⊂{|t|≤1},and|k|≤1, the support of the measures ⊂{||≤ } ∗ − − ζk(ω) t 1 .Letνk(ω, A)=δnk+γk(ω) ζk(ω)(A)=ζk(ω)(A nk γk(ω)). rβ rβ rβ Setting B =sup n 2−|k| ,if|γ | (B +2c)2|k| − n − γ ⊂ k k k k k rβ |t| >c2|k| . Thus, for large enough |k|, | | |k|rβ ≤ | | |k|rβ EΩνk( t > (B +2c)2 ) EΩ ζk t >c2

|k|rβ |k|rβ + P (|γk| >c2 )=P (|γk| >c2 ). Now, by Proposition 2.4, assumption (b) and following the same ideas in Theorem ∞ h h+1 3.3, there exists 0

∈ 1 | ω |≤ C2(ω) By assumption (a), there exist 0 22rβh 2 |t|>22rβh h j∈Ih h ≤ 2 2 C2 (ω)c(β) f 2. Now let D˜ ω = Dω if m ≤ n

Combining the variational inequalities for {Enf} and for the centered averages { ω − } Fn f Enf , the theorem is proven. The key element in the above theorem, is the control of the decay of the averaging kernels in condition (d). Thus, Theorem 4.1 is a particular case of item (c) in the following more general result.

Proposition 4.2. Let {νk}k∈Nr be a sequence of transition measures satisfying the assumptions of Theorem 3.3.LetKn,En and Dn as deﬁned in Section 3. If | |α| ω |∈ 1 there exist α>0 such that supn sup|t|≥1 t Dn (t) L (Ω), then there exists 0 1, ∞ 2 |Dρn f| 1, {En}n∈I(ρ) is variationally good in L then {Kn}n∈I(ρ) is uniformly variationally good in L2. { } 2 { } (b) If En is universally good in L , limρ→1 supn bρn+1 /bρn =1and either νk is a sequence of real–valued positive transition measures or {|En|} is also 2 2 universally good in L ,then{Kn} is uniformly universally good in L . r (c) Assume the total variation |νk(ω)| =1for all k ∈ N and all ω ∈ Ω,thenif 2 {En} is variationally good in L then {Kn} is uniformly variationally good in L2. RANDOMLY PERTURBED ERGODIC AVERAGES 241

Proof. Proving the L2 bound for the square function follows the same ideas as in the proof of the square function in Theorem 3.3 together with the estimate for the second term in Theorem 4.1. Parts (a) and (b) follow as in Theorem 3.3 and (c) follows the same estimates as in the proof of Theorem 4.1. Bounded transition measures with the conditions in Proposition 4.2 also are uniform norm summable. Such result requires Moricz’s [29] estimate for moments of sums of random variables and its extension in Cohen and Cuny [16].

2 Proposition 4.3 ([16]). Let (Y,C,μ) aprobabilityspaceand{Gn}⊂L (Y ).Let {αn} a sequence of non–negative numbers, η>0, C>0 constants, and {Hn} a η non-decreasing sequence with growth condition Hn n such that, for all m>n≥ 1, 2 m m ≤ Gk Hm αk. k=n+1 2 k=n+1 2 ∞ 2 If n≥1 αnHn(log n) < ,then n≥1 Gn converges a.e. in L (Y ),and n 2 2 sup Gk ≤ C(η) αnHn(log n) . n≥1 k=1 2 n≥1

Theorem 4.4. Given a sequence of bounded transition measures {νk}k∈Nr such that, for some β ∈ (0, 1), | | | | |k|rβ ∞ | |α|ω |∈ ∞ EΩ νk ( t >c2 ) < and M(ω)=supsup t νn (t) L (Ω) n | |≥ k∈Nr t 1 2 for some α>0.Then{νk} is uniformly norm summable in L . r Proof. Let In,m = {k ∈ N : n<|k|≤m}. 2 2 − − νkf(x) EΩνkf(x)) ≤ νk(t) EΩνk(t) r r dμf . |k| Rd |k| k∈I k∈I n,m 2 n,m

By Proposition 2.4 and the tail distribution condition on the νk’s, there exists ∞ mrβ 0

rβ On the other hand, for |t| > 2m , ⎡ ⎤ 2 2 − 2 νk(t) EΩνk(t) M(ω) ⎣ 1 ⎦ ≤ (log m) M(ω) rβ . |k|r |t|2α |k|r 22αm k∈In,m k∈In,m Combining both estimates, there exists 0

Proof of Theorem 1.12. It follows from Theorem 4.1 and Theorem 4.4, with estimates for the kernels as in Theorem 4.1.

Acknowledgment We thank the referee for very helpful comments and remarks.

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Division of Mathematics and Computer Science (DMACS), College of Natural and Applied Sciences (CNAS), University of Guam (UOG), ALS 319, UOG Station, Mangilao, 96923 Guam Email address: [email protected] Department of Mathematics and Statistics, University at Albany, SUNY, Albany, New York 12222 Email address: [email protected]