Limit theory for some positive stationary processes with infinite mean

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www.imstat.org/aihp 2014, Vol. 50, No. 1, 256–284

DOI:10.1214/12-AIHP513

Limit theory for some positive stationary processes with infinite mean

Jon Aaronson

^a,1

and Roland Zweimüller

^b

aSchool of Math. Sciences, Tel Aviv University, 69978 Tel Aviv, Israel. E-mail:[email protected] bFaculty of Mathematics, University of Vienna, 1090 Vienna, Austria. E-mail:[email protected]

Received 20 September 2011; revised 16 June 2012; accepted 4 July 2012

Abstract. We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.

Résumé. Nous prouvons des théorèmes limites et des lois du logarithme itéré unilatérales pour une classe de processus stochas- tiques positifs, mélangeants et stationnaires. Cette classe contient en particulier les processus obtenus par des observables noninté- grables de certaines applications dilatantes. Ceci est obtenu en généralisant la théorie de Darling–Kac à une famille appropriée de transformations préservant la mesure.

MSC:Primary 60Fxx; secondary 37A40; 60G10

Keywords:Infinite invariant measure; Transfer operator; Infinite ergodic theory; Darling–Kac theorem; Pointwise dual ergodic; Mixing coefficient; Stable limit; One-sided law of iterated logarithm

Overview

We prove limit theorems for positive, stationary, processes with infinite mean satisfying mixing conditions which occur naturally in certain dynamical systems:stable limit theoremsfor certainϑ_μ-mixing processes andone-sided laws of the iterated logarithmfor certainψ^∗-mixing processes (definitions below).

The method of proof is byinversionwhich is done by first building aKakutani towerover the generating probability preserving transformation, using the time zero observation as height function.

The mixing properties of the stationary process ensure that the resulting infinite measure preserving transformation isweakly pointwise dual ergodic, which allows us to develop a generalizedDarling–Kac theoryfor ergodic sums of this system. The results for the original stochastic process then follow by a standard inversion argument.

We illustrate both the finite-measure and the infinite-measure results by applying them to certain one-dimensional dynamical systems.

1Partially supported by Israel Science Foundation Grant No. 1114/08.

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1. Definitions and preliminaries Stationary processes

We are going to consider partial sums of ergodicR₊-valued stationary processes(ξ_n)_n_≥₀ withE(ξ_n)= ∞. Such a process can always be represented asξn=ϕ◦Sⁿ, whereS is a measure preserving transformation on a probability space(Ω,A, P ), andϕ:Ω→R+is measurable withE(ϕ)= ∞(and w.l.o.g.A=σ{ϕ◦Sⁿ:n≥0}). Due to nonin- tegrability, it will suffice to restrict attention to theN-valued case, as by the ergodic theorem the partial sums of the fractional parts are asymptotically negligible compared to the partial sums of the process. Henceforth,ϕ:Ω→N, and we letα=α_ϕ= {[ϕ=l]:l∈N}. This gives aprobability preserving fibred system(Ω,A, P , S, α)in the sense of the following definition.

Nonsingular transformations and fibred systems

A measurable mapSon aσ-finite space(Ω,A, m)is callednonsingularifm◦S⁻¹m. Itstransfer operator(with respect tom) is the positive linear mapS:L¹(m)→L¹(m)defined by

A

Sfdm=

S−1A

fdm

f∈L¹(m), A∈A .

Afibred or piecewise invertible systemis a quintuple(Ω,A, m, S, α) whereS is a nonsingular transformation on (Ω,A, m), andα⊂Ais a countable, unilateral generator so that the restrictionS:a→Sais invertible, nonsingular on eacha∈α. In this case, for everyk≥1,(Ω,A, m, S^k, αk)is a fibred system, whereαk:=k−1

j=0S⁻^jα.

The transfer operator of(Ω,A, m, S, α)can be represented as Sf =

a∈α

1Sav_a(f◦v_a),

wherev_a:Sa→adenotes the inverse ofS:a→Sa, andv_a :=^dm_dm^◦^v^a.

If m actually is an S-invariant probability measure, the system is called probability preserving, and we write P :=m.

Mixing

LetP(Ω,A)denote the collection of probability measures on(Ω,A), and callμ∈P(Ω,A)equivalenttoP,μ∼P, ifμP μ. Various mixing conditions for a probability preserving fibred system(Ω,A, P , S, α)are defined in terms of the asymptotics of certainmixing coefficients, all defined forn≥1 (and only using pairs of sets for which the denominator is nonzero). The system is called

• ϑ_μ-mixing(for someμ∈P(Ω,A)withμ∼P) ifϑ_μ(n)→0, where ϑ_μ(n):=sup

|P (A∩S⁻⁽ⁿ⁺^k)B)−P (A)P (B)|

μ(B) : k≥1, A∈σ (α_k), B∈A ;

• ψ^∗-mixingifψ^∗(n)→1, where ψ^∗(n):=sup

P (A∩S⁻⁽ⁿ⁺^k)B)

P (A)P (B) : k≥1, A∈σ (α_k), B∈A ;

• ψ-mixingifψ (n)→0, where ψ (n):=sup

|P (A∩S⁻⁽ⁿ⁺^k)B)−P (A)P (B)|

P (A)P (B) : k≥1, A∈σ (α_k), B∈A ; andcontinued fraction mixingif, in addition toψ-mixing,ψ (1) <∞.

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Remark 1. (a)The notion ofϑ_μ-mixing generalizes that of reverseφ-mixingwhich requires thatφ₋(n)→0,where φ₋(n):=sup

|P (A∩S⁻⁽ⁿ⁺^k)B)−P (A)P (B)|

P (B) : k≥1, A∈σ (α_k), B∈A , so thatϑ_P(n)=φ₋(n)for alln.

(b)As shown in [11],ψ^∗(1) <∞implies ψ^∗-mixing.Elementary computation shows that φ₋(n)≤ψ^∗(n)−1, henceψ^∗-mixing entails reverseφ-mixing,that is,ϑ_P-mixing.

(c)It is immediate thatψ-mixing impliesψ^∗-mixing.Moreover,note thatψ^∗(1)≤1+ψ (1).In view of(a),ψ (1) <

∞also impliesψ^∗-mixing.

(d)For examples withψ^∗(1) <∞which are notψ-mixing,see chapter5in[12].

(e)In Section 7below,we consider a class of interval maps(weakly mixing Rychlik-maps)for whichϑ_μ(n)→ 0 exponentially fast(as shown in [8]).We prove that,under some natural extra assumptions,ψ^∗(1) <∞ implies continued fraction mixing in this setup.

Strong distributional convergence and limit laws

For(X,B, m)aσ-finite measure space,F_n:X→ [0,∞]measurable, andY a random variable taking values in[0,∞], we say that(F_n)converges strongly in distribution toY, written

F_n−→^d

n→∞Y,

if it converges in law with respect to all absolutely continuous probabilities, that is, if for all continuous (hence bounded)g:[0,∞] →Rand allP ∈P(X,B)withP m,

X

g(F_n)dP −→

n→∞E g(Y )

.

Forγ ∈ [0,1]we letY_γ ≥0 denote a random variable which has the normalized Mittag-Leffler distribution of orderγ, that is,E(Y_γ^p)=^p^!₍₁⁽⁽¹₊⁺_{pγ )}^{γ ))}^p forp≥0. EvidentlyY1≡1, andY0has exponential distribution. Also,Y1/2is the absolute value of a centered Gaussian random variable.

Forγ∈(0,1], the variableZ_γ :=Y⁻^1/γ then has apositiveγ-stable distributionwithE(e⁻^{t Z}^γ)=exp(−(1+ γ )t^γ)fort >0.

2. Results on stationary processes

Recall that a functiona:(L,∞)→(0,∞)isregularly varying of indexρ∈Rif it is measurable anda(ct )/a(t )→c^ρ ast→ ∞for allc >0. In caseρ >0, such ana has anasymptotically inverse functionb (uniquely determined up to asymptotic equivalence), meaning thata(b(t ))∼b(a(t))∼t ast→ ∞, andbis regularly varying of index 1/ρ.

These concepts are extended to sequences(a(n))by interpreting them as functions onR₊ viat−→a(t)(which we shall do without further mention).

In the statements below,(Ω,A, P , S, α)is a probability preserving fibred system, andϕ:Ω→Nisα-measurable.

We let ϕn:=

n−1

k=0

ϕ◦S^k, n≥0,

denote the partial sums of the stationary process(ξ_n)_n_≥₀=(ϕ◦Sⁿ)_n_≥₀, and define aϕ(n):=

n

k=1

P

[ϕk≤n]

, n≥0.

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In order to establish our results, we will need to assume that the growth of a_ϕ(n) is adapted to the decay of the mixing coefficients of the process. The main condition is as follows although we will use a stronger version (2.6) in Theorem2.3.

Definition of adaptedness

Letτ (n)↓0. We will say that the sequence(a(n))_n_∈Nin(0,∞)isadapted to(τ (n))_n_∈Nif nτ (δa(n))

a(n) −→

n→∞0 for allδ >0. (2.1)

Observe that in this case any sequence asymptotically equivalent to(a(n))is adapted, too. Note also that if(a(n))is regularly varying with positive index, then it is adapted to(τ (n))as soon as (2.1) holds for oneδ >0.

Our first result is a distributional limit theorem. In the barely infinite measure case (γ =1) it comes with an associated a.e. result. For γ ∈(0,1), corresponding statements will be established under stronger assumptions in Theorem2.3below.

Theorem 2.1. Suppose that(Ω,A, P , S, α)is aϑ_μ-mixing probability preserving fibred system,and thatϕ:Ω→N isα-measurable.

(a) (Stable limit theorem)Let(a(n))=(aϕ(n))beγ-regularly varying for someγ ∈(0,1],and assume that it is adapted to(ϑ_μ(n)).Then,

ϕ_n b(n)

−→d

n→∞Z_γ, (2.2)

wherebis asymptotically inverse toa,b(a(n))∼a(b(n))∼n.

(b) (One-sided law of the iterated logarithm forγ =1)If,in addition,γ =1andb(n/log logn)log logn∼b(n), then

lim

n→∞

ϕn

b(n)=1 a.s. (2.3)

Remark 2. (a)The conclusion of Theorem2.1(a) was established for certain φ-mixing processes in([17], Corol- lary5.10)and for continued fraction mixing processes in[13] (see also[3]).

(b)A functional version of (a)is also valid,and can be proved using a straightforward,appropriate adaptation of[9].

(c)An analogue of Theorem2.1(b)forψ-mixing processes was established in[4].

The results mentioned in Remark2(a) also compute thea_ϕ(n)from the marginal distributions, for which additional

“close correlation” assumptions such asψ^∗(1) <∞are required. We now show how to determine the asymptotics of a_ϕ(n)from the marginal distributions under the weaker close correlation condition (2.4) (but we still use the stronger ψ^∗(1) <∞in Theorem2.3below).

Theorem 2.2 (Identifying the normalization). Let (Ω,A, P , S, α)be a ϑ_μ-mixing probability preserving fibred system.Assume thatϕ:Ω→Nisα-measurable,and that there exists someΦ∈L¹(P )₊such that

S(ϕ∧n)≤E(ϕ∧n)Φ ∀n≥1. (2.4)

Assume that(a(n))˜ is regularly varying with indexγ∈(0,1],and adapted to(ϑμ(n)),then E(ϕ∧n) ∼

n→∞

n

(2−γ )(1+γ )a(n)˜ (2.5)

implies

aϕ(n) ∼

n→∞a(n).˜

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Finally, replacing adaptedness (2.1) to (ϑ_μ(n)) by the stronger assumption (2.6) involving a(a(n)) below, we establish the following pointwise result.

Theorem 2.3 (The one-sided law of the iterated logarithm). Suppose that(Ω,A, P , S, α)is aψ^∗-mixing proba- bility preserving fibred system withψ^∗(1) <∞,and thatϕ:Ω→Nisα-measurable.

If(a(n))=(a_ϕ(n))isγ-regularly varying for someγ∈(0,1)with asymptotic inverseb(n),and if nϑP(δa(a(n)))

a(n) −→

n→∞0 for someδ >0, (2.6)

then,lettingCγ :=Kγ⁻^1/γ withKγ:=_γγ⁽¹(1−⁺γ )^{γ )}¹⁻^γ,we have lim

n→∞

ϕ_n

b(n/log log(n))log log(n)=C_γ a.s. (2.7)

Moreover,if(τ (n))is any sequence in(0,∞)withτ (n)↑andτ (n)/n↓,letκτ be the unique number in[0,∞]such that

n≥1e⁻^{κτ (n)}/nconverges forκ > κτ and diverges forκ < κτ.Then lim

n→∞

ϕ_n

b(n/κτ (n))κτ (n)≥C_γ a.s.ifκ∈ [κ_τ,∞), (2.8)

and lim

n→∞

ϕ_n

b(n/κτ (n))κτ (n)≤C_γ a.s.ifκ∈(0, κτ]. (2.9)

Remark 3. (a)The conclusion of Theorem2.3was established for iid processes in[21],and forψ-mixing processes in [5].Our proof of Theorem2.3is by establishing the conditions needed for the methods of[5].Therefore the functional version also follows as in[6].

(b)Recall thatϑ_P(n)=φ₋(n),see Remark1(a).

Inversion:Kakutani towers and return time processes

Our results will be established using the well-known technique of “inverting” corresponding results for infinite measure preserving transformations, the connection being established via the following concept. Given a probability preserving transformation(Ω,A, P , S)and a measurable functionϕ:Ω→N, theKakutani towerof(Ω,A, P , S, ϕ)is the object(X,B, m, T )with(X,B, m)theσ-finite space defined by

• X:=

n≥1[ϕ≥n] × {n},

• B:= {

n≥1Bn× {n}: Bn∈A∩ [ϕ≥n] ∀n≥1},

• m(A× {n}):=P (A),

andT:X→Xis the map given by

• T (x, n):=

(x, n+1), ϕ(x) > n, (Sx,1), ϕ(x)=n.

It follows that(X,B, m, T )is a conservative, measure preserving transformation which is ergodic iff(Ω,A, P , S)is ergodic.

This “tower building process” is reversible. Given a conservative ergodic σ-finite measure preserving system (X,B, m, T ), we define thereturn time process of T onΩ ∈F:= {B∈B: 0< m(B) <∞}as theN-valued stationary process(ϕ_Ω◦T_Ωⁿ)_n_≥₀on(Ω,B∩Ω, m_Ω, ), where

• ϕ(x)=ϕΩ(x):=min{n≥1:Tⁿx∈Ω},

• TΩ(x):=T^ϕ(x)(x), and

• mΩ(A):=m(A∩Ω)/m(Ω).

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It follows that the Kakutani tower of(Ω,B∩Ω, m_Ω, T_Ω, ϕ_Ω)is a factor of(X,B, m, T )wherem=_m(Ω)¹ m(and an isomorph in caseT is invertible).

Now setϕj:=j−1

i=0ϕΩ◦T_Ωⁱ, which is the time of thenth return toΩ. It is straightforward to check that these are dual to theoccupation timesofΩ,S_n(1Ω):=_n₋1

k=01Ω◦T^kin that

S_n(1Ω)≤j iff ϕ_j≥n. (2.10)

This entails, via routine arguments, that various properties of (ϕ_j)_j_≥₁are equivalent to corresponding properties of (S_n(1Ω))_n_≥₁. Specifically, suppose thata(n)isγ-regularly varying withγ∈(0,1], and letbbe asymptotically inverse toa. Then, forY a[0,∞]-valued random variable,

1

a(n)S_n(1Ω)−→^d m(Ω)Y iff ϕ_n b(n)

−→d

1 m(Ω)Y

1/γ

, (2.11)

and

nlim→∞

1

a(n)S_n(1Ω)^a.e.=m(Ω) iff lim

n→∞

ϕ_n b(n)

a.e.= 1

m(Ω) 1/γ

. (2.12)

3. Weak pointwise dual ergodic measure preserving transformations

In this section, we consider the properties of infinite ergodic systems to be used in the proofs of the above results.

Weak pointwise dual ergodicity

LetT be a conservative, ergodic, measure preserving transformation (not necessarily invertible) on theσ-finite space (X,B, m), andT:L¹(m)→L¹(m)its transfer operator, which naturally extends to all nonnegative measurable functions. Invariance ofmmeans thatT1X=1X, and sinceT is conservative ergodic, any measurableg:X→ [0,∞) which is subinvariant,T g ≤g, is, in fact, constant. Hurewicz’s ratio ergodic theorem (Theorem 2.2.1 of [1]), guaran- tees that

_n₋₁

k=0T^kf n−1

k=0T^kg −→

n→∞

m(f )

m(g) a.e. onX (3.1)

for allf, g∈L¹₊(m):= {f ∈L¹(m): f ≥0 andm(f ) >0}. (Due to conservativity,_n₋₁

k=0T^kf → ∞a.e. for such f.)

Throughout, convergence in measure, −→^m , for our σ-finite measure m, is understood to mean convergence in measure,−→, for every finite^ν νm(or, equivalently, for allν=m_AwithA∈F).

The conservative ergodic measure preserving transformation T on the σ-finite space (X,B, m) will be called weakly pointwise dual ergodicif there exist constantsa(n) >0, n≥1,such that

1 a(n)

n−1

k=0

T^kf −→^m

n→∞

X

fdm forf ∈L¹₊(m), (3.2)

and

nlim→∞

1 a(n)

n−1

k=0

T^kf =

X

fdm a.e. onXforf ∈L¹₊(m). (3.3)

This generalizes the notion ofpointwise dual ergodicity(cf. Section 3.7 of [1], or [2]), which requiresa(n) >0 such that

1 a(n)

n−1

k=0

T^kf −→

n→∞

X

fdm a.e. onXforf ∈L¹₊(m). (3.4)

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In either case the return sequence (a(n))_n_≥₁, which is determined up to asymptotic equivalence and satisfies a(n+1)∼a(n)→ ∞, can (and will) be taken nondecreasing. It is usually denoted(an(T ))n≥1.

Remark 4. No invertibleconservative ergodic measure preserving transformation(X,B, m, T )withm(X)= ∞is pointwise dual ergodic. (Since in this caseT f =f ◦T⁻¹,so that(3.4)would give a pointwise ergodic theorem with normalizing constantsa(n)forT⁻¹,which is impossible,see Section2.4of[1].)However,invertible systems can still be weakly pointwise dual ergodic.

Example 1. For a concrete example,letT:(0,1)→(0,1)be given byT x:=₁₋^x_x forx < ¹₂ andT x:=2x−1for x >¹₂,which is conservative ergodic with invariant measuremhaving density ¹_x,and definea(n):=n/logn.By the Darling–Kac theorem for pointwise dual ergodic transformations, (see[2,14],Section3.7of[1],or[24]),

1 a(n)

n−1

k=0

f◦T^k −→^m

n→∞

X

fdm forf ∈L¹₊(m), and according to Proposition2of[4],

nlim→∞

1 a(n)

n−1

k=0

f ◦T^k=

X

fdm a.e.onXforf ∈L¹₊(m).

These carry over to the natural extensionT_∗ ofT:This is immediate for functions of the formf_∗=f ◦π,withπ denoting the factor map.Then use the ratio ergodic theorem to pass to generalf_∗∈L¹₊(m_∗).Therefore,the invertible conservative ergodic measure preserving transformationT_∗⁻¹,with transfer operatorf →f◦T_∗is weakly pointwise dual ergodic,and hence so isT_∗,withan(T_∗)∼an(T_∗⁻¹)∼a(n).

Conditions of this flavour can be exploited most efficiently if one succeeds in identifying special sets on which there is additional control on the convergence. Recall (cf. [1,19]) thatA∈F= {B∈B: 0< m(B) <∞}is called a uniform set(forf ∈L¹₊(m)) if there area(n) >0, such that

ess sup

A

1 a(n)

n−1

k=0

T^kf −m(f ) −→

n→∞0, and, more specifically, aDarling–Kac setif

ess sup

A

1 a(n)

n−1

k=0

T^k1A−m(A) −→

n→∞0.

We now defineA∈Fto be alimited setif there exist constantsa(n) >0 such that 1

a(n)

n−1

k=0

T^k1A m_A

n−→→∞m(A), (3.5)

and

ess sup

A

1 a(n)

n−1

k=0

T^k1A−→

n→∞m(A). (3.6)

Due to (3.1) the asymptotics of(a(n))does not depend onf orA. It is not hard to see that defining aA(n):=

n−1

k=0

m(A∩T⁻^kA)

m(A)² forA∈F,

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we have a(n) ∼

n→∞a_A(n) in (3.5) and (3.6) for every limited setA. (3.7)

The existence of uniform sets is equivalent to pointwise dual ergodicity, anda(n)∼an(T )in this case. (But we do not know if every pointwise dual ergodic transformation has a Darling–Kac set.) Similarly,weak pointwise dual ergodicity is equivalent to the existence of limited sets, as we have

Proposition 3.1 (Limited sets and weak pointwise dual ergodicity from local behaviour). LetT be a conservative ergodic measure preserving transformation on theσ-finite space(X,B, m).

(a)Suppose there areA∈F,f∈L¹₊(m),and constantsa(n) >0,n≥1,such that 1

a(n)

n−1

k=0

T^kf −→^m^A

n→∞

X

fdm, (3.8)

and

nlim→∞

1 a(n)

n−1

k=0

T^kf =

X

fdm a.e.onA. (3.9)

Then,for everyη >0,T possesses a limited setA∈F∩Awith return sequence(a(n))n≥1andm(A)≥m(A)−η.

(b)IfT has a limited set,then it is weakly pointwise dual ergodic.

Proof. (a) LetB₀:=A. By Hurewicz’s ratio ergodic theorem we may assume w.l.o.g. thatf =1B0, and given any setB_j∈F∩Awithm(B_j) > m(A)−η, we also havea(n)⁻¹_n₋₁

k=0T^k1Bj →m(B_j)a.e. onA. Egorov’s theorem then provides us with someBj+1∈F∩Bj such thatm(Bj+1) > m(A)−η, and

1 a(n)

n−1

k=0

T^k1B_j −→

n→∞m(Bj) uniformly onBj+1.

Using this to inductively define a decreasing sequence(B_j)_j_≥₀inF, we obtain a setA:=

j≥0B_j withm(A)≥ m(A)−η. Givenε >0 choosej withm(Bj) < m(A)+ε/2. Then, forn≥nj(ε),

ess sup

A

1 a(n)

n−1

k=0

T^k1A≤ess sup

Bj+1

1 a(n)

n−1

k=0

T^k1Bj < m(B_j)+ε 2.

This gives the required control from above. Since, by (3.1), (3.8) also holds withf =1A, we see thatAis indeed a limited set.

(b) Now start from the assumption thatT has a limited setA. Due to Hurewicz’s theorem, weak pointwise dual ergodicity follows as soon as we check the defining conditions (3.2) and (3.3) forf=1A, i.e. we have to prove

fn:= 1 a(n)

n−1

k=0

T^k1A

−→m

n→∞m(A), (3.10)

and

f¯:= lim

n→∞

1 a(n)

n−1

k=0

T^k1A=m(A) a.e. onX. (3.11)

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LettingA_n:=A∩ [ϕ_A> n],n≥0, we have, by routine arguments,

n≥0

Tⁿ1An=1X, (3.12)

and decomposing, for anyn≥0, A according to the time of the last return before time n, A=A_n∪_n₋₁

k=0A∩ T⁻⁽ⁿ⁻^k)(Ak)(disjoint), we find that forN≥0,

N

n=0

Tⁿ1A=

N−1

k=0

T^k

1A_k N−k

j=0

T^j1A

+

N

n=0

Tⁿ1A_n. (3.13)

SinceAis a limited set, there is someM∈(0,∞)such that 1

a(n)

n−1

k=0

T^k1A≤M onAforn≥1. (3.14)

With (3.14) and (3.12) providing bounds for the sums in (3.13), we get 1

a(N )

N−1

n=0

Tⁿ1A≤M·1X forN≥1, (3.15)

whereM:=M+1/a(1)∈(0,∞). Consequently,f¯≤M·1X. Now observe that|T f _n−f_n| ≤2/a(n)→0. Com- bined with a variant of Fatou’s lemma forT, Lemma3.1below, this yields

Tf¯≥ lim

n→∞T f _n= ¯f a.e. onX.

Henceg:=M·1X− ¯f≥0 is subinvariant,T g ≤g. AsT is conservative ergodic, this implies thatgis constant a.e., and hence so isf¯. In view of (3.5), this yields (3.11).

To finally prove convergence in measure onX, it suffices to check it on each of the setsT⁻^lA,l≥0, since these coverX. But for eachl,

T⁻^lA

1 a(n)

n−1

k=0

T^k1A

dm=a_A(n+l)−a_A(l)

a(n) m(A)−→

n→∞m(A).

Together with (3.15) and (3.11) this entailsa(n)⁻¹n−1 k=0T^k1A

m_{T−l A}

−→m(A), and hence (3.10).

The preceding proof made use of

Lemma 3.1. LetT be measure preserving on theσ-finite space (X,A, m),with transfer operatorT,and letM∈ (0,∞).

(a)Ifg, gk:X→ [0, M]are measurable withgk≥gk+1ga.e.,then

T g_kT g a.e.onX. (3.16)

(b)Iff_n:X→ [0, M]are measurable functions,then T (lim

n→∞f_n)≥ lim

n→∞T f _n a.e.onX. (3.17)

Proof. (a) By positivity of T, we haveT M≥T g 1≥T g 2≥ · · · ≥T g a.e., whileT M=M by invariance ofm.

Now take anyA∈A,m(A) <∞, then 1T−1Ag_k−→1_T−1AginL₁(m). Due toL₁-continuity ofT, this entails 1AT g _k=T (1 _T−1Ag_k)−→T (1 _T−1Ag)=1AT g inL₁(m),

and sinceAwas arbitrary, the latter implies (3.16).

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(b) Letg_k:=sup_n_≥_kf_n≥f_k, andg:=limn→∞f_n, theng, g_k satisfy the assumptions of (a). Hence,T g _kT g a.e. By positivity, however, we haveT g_k≥T f _k a.e. for allk≥1. Together, these observations give (3.17).

The notion of return sequences has originally been introduced in the context of rationally ergodic transformations (cf. Section 3.3 of [1]). The present use of the term is justified by

Proposition 3.2 (Weak pointwise dual ergodicity implies rational ergodicity). LetT be a weakly pointwise dual ergodic measure preserving transformation on theσ-finite space(X,B, m).Then every limited setAsatisfies a Rényi inequality,meaning that there is someM=M(A)∈(0,∞)such that

A

S_n(1A)2

dm≤M

A

S_n(1A)dm 2

forn≥1.

In particular,T is rationally ergodic with(a_n(T ))_n_≥₁a return sequence in the sense of[1].

Proof. Same as the proof of Proposition 3.7.1 in [1], using (3.7) and the existence of limited sets established above.

(Weak)pointwise dual ergodicity and special sets for Kakutani towers

We are going to show that Kakutani towers aboveϑμ-mixing systems satisfying our adaptedness conditions are weakly pointwise dual ergodic. Moreover, in the presence of regular variation,ϑμ-mixing with sufficiently fast rate implies pointwise dual ergodicity.

Note first that if(X,B, m, T )is a conservative ergodic measure preserving transformation andΩ∈B, m(Ω)=1, has return timeϕ=ϕΩ, andϕj:=j−1

i=0ϕ◦T_Ωⁱ , then, in view of (2.11), a_Ω(n)∼

n

k=1

m

Ω∩T⁻^kΩ

= n

j=1

m

Ω∩ [ϕ_j≤n]

=a_ϕ(n). (3.18)

Via (3.7) this links(a_ϕ(n))to the asymptotic type of a weakly pointwise dual ergodic system.

Theorem 3.1 ((Weak) pointwise dual ergodicity viaϑ_μ-mixing return processes). Let(X,B, m, T )be a conser- vative ergodic measure preserving transformation on a σ-finite space,and suppose thatΩ ∈B, m(Ω)=1,has a countable partitionα⊂B∩Ω such thatϕ=ϕ_Ω isα-measurable and that(Ω,B∩Ω, P , T_Ω, α)isϑ_μ-mixing for someμequivalent toP :=m_Ω.

(a)If(a(n)):=(a_Ω(n))is adapted to(ϑ_μ(n)),thenT is weakly pointwise dual ergodic witha_n(T )∼a(n).In addition,∀ε >0 ∃N_εs.t.∀n≥N_ε,

1 a(n)

n

k=1

T^k1Ω−1≤ε

1+dμ dP

a.e.onΩ. (3.19)

In particular,if_dP^dμ_∞<∞,thenΩis a limited set.

(b)If(a(n))_n_≥₁isγ-regularly varying for someγ∈(0,1]and ϑ_μ(n)=O

1 n^r

for somer > 1 γ −1,

then(a(n))is adapted to(ϑ_μ(n)),andT is pointwise dual ergodic witha_n(T )∼a(n).

(c)If(a(n))_n_≥1isγ-regularly varying for someγ∈(0,1],andψ^∗(1) <∞,and nϑ_μ(δa(a(n)))

a(n) −→

n→∞0 for allδ >0, (3.20)

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then(a(n))is adapted to(ϑ_μ(n)),andT is pointwise dual ergodic witha_n(T )∼a(n).Moreover,∀ε >0 ∃N_ε such that∀n≥Nε

1 a(n)

n

k=1

T^k1Ω−1 ≤ε

1+dμ

dP

a.e.onΩ. (3.21)

In particular,if^dμ_dP∞<∞,thenΩ is a Darling–Kac set.

Proof. We write(Ω,A, P , S, α):=(Ω,B∩Ω, m_Ω, T_Ω, α). Note first that by definition ofϑ_μ, we have forj, p≥1 andA∈αj,

B

S^j⁺^p1AdP ≤

B

P (A)+ϑ_μ(p)dμ dP

dP for arbitraryB∈A, and hence

S^j⁺^p1A≤P (A)+ϑ_μ(p)dμ

dP a.e. onΩ. (3.22)

We claim that forn, p≥1, T_n:=

n

k=1

T^k1Ω≤p+a(n)+nϑ_μ(p)dμ

dP a.e. onΩ. (3.23)

To see this, observe thatT^k1Ω=

j≥1S^j1_[ϕ_j=k]a.e. onΩ, and hence

T_n= n

j=1

S^j1_[ϕ_j≤n]≤

n+p

j=1

S^j1_[ϕ_j≤n]

≤p+ n

j=1

S^j⁺^p1_[ϕj+p≤n]≤p+ n

j=1

S^j⁺^p1_[ϕj≤n]

≤p+ n

j=1

P

[ϕ_j≤n]

+ϑ_μ(p)dμ dP

=p+a(n)+nϑ_μ(p)dμ

dP a.e. onΩ, since[ϕj≤n] ∈αj.

Proof of (a): To establish (3.19), we letp=p_n,ε:= εa(n)forε >0. By adaptedness we can chooseN_ε such thatnϑ_μ(p_n,ε) < εa(n)whenevern≥N_ε. Then by (3.23),

T_n≤a(n)

1+ε

1+dμ dP

a.e. onΩforn≥N_ε, (3.24)

as required.

To prove weak pointwise dual ergodicity, we will use part (a) of Proposition3.1. WriteR_n:=T_n/a(n)and observe first that by (3.19),

nlim→∞R_n≤1 a.e. onΩ. (3.25)

In fact, (3.19) shows that for anyM >0, the estimate (3.25) holds uniformly onΩ∩ [_dP^dμ≤M]. Consequently, P

R_n≥1+ε

n−→→∞0 for everyε>0. (3.26)

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To obtain the analogous statement for convergence from below, we observe that fort∈(0,1), R_n≤1+ε

1+dμ

dP

a.e. onΩ ⇒ P

[R_n≤t]

≤ 2ε

1+ε−t, (3.27)

since (Edenoting expectation with respect toP) 1=E(Rn)=E(Rn1_[R_n>t])+E(Rn1_[R_n≤t])

≤(1+ε)P

[Rn> t] +εμ

[Rn> t] +t P

[Rn≤t]

≤1+2ε−(1+ε−t )P

[R_n≤t] .

Now fixε∈(0,1)and takeN_ε as in (3.24). Fort:=1−√

ε∈(0,1), observation (3.27) then yields P

[R_n≤1−√ ε]

≤2√

ε forn≥N_ε. This readily implies

P

R_n≤1−ε

n−→→∞0 for everyε>0,

which, together with (3.26), gives (3.8). Combined with (3.25) the latter yields (3.9). (There is some subsequence n_k ∞such thatR_n_k→1 a.e. forP.)

Proof of (b): Computation of the indices of regular variation shows that the assumptions of (b) entail adaptedness.

As in part (a) this implies (3.25).

To check pointwise dual ergodicity, it remains to show that lim

n→∞Rn≥1 a.e. onΩ. (3.28)

(Proposition 3.7.5 of [1] ensures that a.e. convergence on someA∈Fsuffices.) To this end, choosec≥1 such that ϑ_μ(n)≤c/n^r for alln≥1. There exists >0 andN₀so that

a(n) > n^1/(r⁺¹⁾⁺^2s ∀n > N₀. Choosingp=n^1/(r⁺¹⁾in (3.23) we have

Rn= T_n

a(n) ≤1+ 1 n^2s

1+dμ

dP

onΩ∀n≥N0. Due to (3.27), we then see that fort∈(0,1),

q_t(n):=P

[R_n≤t]

≤2c n^s ·

1−t+ c n^s

₋1

≤ 2c

(1−t )n^s forn≥N₀. Since, for alltandλ >1,

n≥1q_t(λⁿ) <∞, the Borel–Cantelli Lemma now implies lim

n→∞Rλⁿ≥1 a.s. onΩfor allλ >1. (3.29)

To finally prove convergence (3.28) of the full sequence, fix any λ >1 and choose integers κ_n(λ) ∞ so that λ^κⁿ^(λ) ≤n≤ λ^κⁿ^(λ)⁺¹. Then regular variation of(a(n))_n_≥₁yields

T_n

a(n)≥ T_λκn(λ)

a(λ^κⁿ^(λ)⁺¹) ∼

n→∞

1 λ^γ

T_λκn(λ)

a(λ^κⁿ^(λ)) a.e. onΩ.

In view of (3.29), we thus have lim_n_→∞Rn≥λ⁻^γ a.s. onΩfor allλ >1, and (3.28) follows.

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Proof of(c): Note first that adaptedness, and hence (3.19) holds. Thus, to prove (3.21), it suffices to check that for n≥N_ε,

1 a(n)

n

k=1

T^k1Ω≥1−ε

1+dμ dP

a.e. onΩ. (3.30)

We show first that∀p, q, n∈Nwithn≥q, T_n≥a(n)−p−nϑ_μ(p)dμ

dP −ψ^∗(1)² P

[ϕ_p> q]

a(n)+q

(3.31) a.e. onΩ. To see this, observe that

T_n = n

j=1

S^j1_[ϕj≤n]≥ n

j=1

S^j⁺^p1_[ϕj+p≤n]−p

= n

j=1

S^j⁺^p1_[ϕ_j≤n]− n

j=1

S^j⁺^p1_[ϕ_j≤n<ϕ_j₊_p]−p

=:Σ1−Σ2−p.

Now, because[ϕj≤n] ∈αj, we have Σ₁≥

n

j=1

P

[ϕ_j≤n]

−ϑ_μ(p)dμ dP

=a(n)−nϑ_μ(p)dμ dP. On the other hand,

Σ₂≤ψ^∗(1) n

j=1

P

[ϕ_j≤n < ϕ_j₊_p]

=ψ^∗(1) n

j=1

n

=j

P

ϕ_j=, ϕ_p◦S^j> n−

≤ψ^∗(1)² n

j=1

n

=j

P

[ϕj=] P

[ϕp> n−]

=ψ^∗(1)²

Σ₂+Σ₂ with

Σ₂ :=

n

j=1 n−q

=j

P

[ϕ_j=] P

[ϕ_p> n−]

≤ P

[ϕ_p> q]ⁿ

j=1 n−q

=j

P

[ϕ_j=]

≤ P

[ϕ_p> q]ⁿ

j=1

P

[ϕ_j≤n]

=P

[ϕp> q] a(n);