Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

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www.imstat.org/aihp 2015, Vol. 51, No. 2, 545–556


© Association des Publications de l’Institut Henri Poincaré, 2015

Weak convergence to stable Lévy processes for nonuniformly hyperbolic dynamical systems

Ian Melbourne


and Roland Zweimüller


aMathematics Institute, University of Warwick, Coventry, CV4 7AL, UK. E-mail:i.melbourne@warwick.ac.uk bFaculty of Mathematics, University of Vienna, Vienna, Austria. E-mail:roland.zweimueller@univie.ac.at

Received 28 February 2013; revised 20 September 2013; accepted 22 September 2013

Abstract. We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau–Manneville intermittency maps, where convergence for the induced system is in the standard SkorohodJ1topology. For the full system, convergence in theJ1topology fails, but we prove convergence in theM1topology.

Résumé. Nous considérons des principes d’invariance faibles (théorèmes limites fonctionnels) dans le domaine d’une loi stable.

Un résultat général est obtenu en relevant de telles lois limites depuis un système dynamique induit vers le système original. Une classe importante d’exemples couverte par notre résultat est donnée par les transformations intermittentes à la Pomeau–Manneville, où la convergence pour le système induit est dans la topologieJ1de Skorohod standard. Pour le système complet, il n’y a pas de convergence dans la topologieJ1, mais nous prouvons la convergence dans la topologieM1.

MSC:Primary 37D25; secondary 28D05; 37A50; 60F17

Keywords:Nonuniformly hyperbolic systems; Functional limit theorems; Lévy processes; Induced dynamical systems

1. Introduction

For large classes of dynamical systems with good mixing properties, it is possible to obtain strong statistical limit laws such as the central limit theorems and its refinements including the almost sure invariance principle (ASIP) [5, 10,12,15,20,21,25–27]. An immediate consequence of the ASIP is the weak invariance principle (WIP) which is the focus of this paper.

Thus the standard WIP (weak convergence to Brownian motion) holds for general Axiom A diffeomorphisms and flows, and also for nonuniformly hyperbolic maps and flows modelled by Young towers [35,36] with square integrable return time function (including Hénon-like attractors [7], finite horizon Lorentz gases [25], and the Lorenz attractor [22]).

Recently, there has been interest in statistical limit laws for dynamical systems with weaker mixing properties such as those modelled by a Young tower where the return time function is not square integrable. In the borderline case where the return time lies inLp for allp <2, it is often possible to prove a central limit theorem with nonstandard norming (nonstandard domain of attraction of the normal distribution). This includes important examples such as the infinite horizon Lorentz gas [32], the Bunimovich stadium [4] and billiards with cusps [3]. In such cases, it is also possible to obtain the corresponding WIP (see, for example, [3,11]).

1The research of Ian Melbourne was supported in part by EPSRC Grant EP/F031807/1 held at the University of Surrey.


For Young towers with return time function that is not square-integrable, the central limit theorem generally fails.

Gouëzel [18] (see also Zweimüller [37]) obtained definitive results on convergence in distribution to stable laws. The only available results on the corresponding WIP are due to Tyran-Kami´nska [33] who gives necessary and sufficient conditions for weak convergence to the appropriate stable Lévy process in the standard SkorohodJ1topology [31].

However in the situations we are interested in, theJ1topology is too strong and the results in [33] prove that weak convergence fails in this topology.

In this paper, we repair the situation by working with theM1topology (also introduced by Skorohod [31]). In particular, we give general conditions for systems modelled by a Young tower, whereby convergence in distribution to a stable law can be improved to weak convergence in theM1topology to the corresponding Lévy process.

The proof is by inducing (see [19,28,30] for proofs by inducing of convergence in distribution). Young towers by definition have a good inducing system, namely a Gibbs–Markov map (a Markov map with bounded distortion and big images [1]). The results of Tyran-Kami´nska [33] often apply positively for such induced maps (see for example the proof of Theorem4.1below) and yield weak convergence in theJ1topology, and hence theM1topology, for the induced system. The main theoretical result of the present paper discusses howM1convergence in an induced system lifts to the original system (even when convergence in theJ1topology does not lift).

As a special case, we recover the aforementioned results [3,11] on the WIP in the nonstandard domain of attraction of the normal distribution.

In the remainder of theIntroduction, we describe how our results apply to Pomeau–Manneville intermittency maps [29]. In particular, we consider the family of mapsf:XX,X= [0,1], studied by [23], given by

f (x)=

x(1+2γxγ), x∈ [0,12],

2x−1, x(12,1]. (1.1)

Forγ∈ [0,1), there is a unique absolutely continuous ergodic invariant probability measureμ. Suppose thatφ:X→ Ris a Hölder observable with

Xφdμ=0. Letφn=n1

j=0φfj. For the map in (1.1), our main result implies the following:

Theorem 1.1. Letf:[0,1] → [0,1]be the map(1.1)withγ(12,1)and setα=1/γ.Letφ:[0,1] →Rbe a mean zero Hölder observable and suppose thatφ(0)=0.DefineWn(t)=n1/αφnt.ThenWn converges weakly in the SkorohodM1topology to anα-stable Lévy process. (The specific Lévy process is described below.)

Remark 1.2. TheJ1andM1topologies are reviewed in Section2.1.Roughly speaking,the difference is that theM1

topology allows numerous small jumps forWnto accumulate into a large jump forW,whereas theJ1topology would require a large jump forW to be approximated by a single large jump forWn.Since the jumps inWnare bounded by n1/α|φ|,it is evident that in Theorem1.1convergence cannot hold in theJ1topology.

Situations in the probability theory literature where convergence holds in theM1topology but not theJ1topology include[2,6].

Theorem1.1completes the study of weak convergence for the intermittency map (1.1) withγ∈ [0,1)and typical Hölder observables. We recall the previous results in this direction. Ifγ∈ [0,12)then it is well known thatφsatisfies a central limit theorem, son1/2φn converges in distribution to a normal distribution with mean zero and variance σ2, whereσ2is typically positive. Moreover, [25] proved the ASIP. An immediate consequence is the WIP:Wn(t)= n1/2φntconverges weakly to Brownian motion.

Ifγ=12 andφ(0)=0, then Gouëzel [18] proved thatφis in the nonstandard domain of attraction of the normal distribution:(nlogn)1/2φnconverges in distribution to a normal distribution with mean zero and varianceσ2>0.

Dedecker and Merlevede [11] obtained the corresponding WIP in this situation (withWn(t)=(nlogn)1/2φnt).

Finally, ifγ(12,1)andφ(0)=0, then Gouëzel [18] proved thatn1/αφnconverges in distribution to a one-sided stable lawGwith exponentα=γ1. The stable law in question has characteristic function

E eit G



1−i sgn φ(0)t

tan(απ/2) ,

wherec=14h(12)(α|φ(0)|)αΓ (1α)cos(απ/2)andh= dx is the invariant density. Let{W (t );t≥0}denote the correspondingα-stable Lévy process (so{W (t )}has independent and stationary increments with cadlag sample paths


andW (t )=dt1/αG). Tyran-Kami´nska [33] verified thatWn(t)=n1/αφntdoes not converge weakly toW in the J1topology. In contrast, Theorem1.1shows thatWnconverges weakly toWin theM1topology.

The remainder of this paper is organised as follows. In Section2we state our main abstract result, Theorem2.2, on inducing the WIP. In Section3 we prove Theorem2.2. In Section4we consider some examples which include Theorem1.1as a special case.

2. Inducing a weak invariance principle

In this section, we formulate our main abstract result Theorem2.2. The result is stated in Section2.2after some preliminaries in Section2.1.

2.1. Preliminaries

Distributional convergence. To fix notations, let(X, P ) be a probability space and (Rn)n1 a sequence of Borel measurable mapsRn:XS, where(S, d)is a separable metric space. Then distributional convergence of(Rn)n1

w.r.t.P to some random elementRofSwill be denoted byRnP R.Strong distributional convergenceRnL(μ)R on a measure space(X, μ)means thatRnP Rfor all probability measuresP μ.

Skorohod spaces. We briefly review the required background material on the SkorohodJ1andM1topologies [31]

on the linear spacesD[0, T],D[T1, T2], andD[0,)of real-valuedcadlagfunctions (right-continuousg(t+)=g(t) with left-hand limits g(t)) on the respective interval, referring to [34] for proofs and further information. Both topologies are Polish, withJ1stronger thanM1.

It is customary to first deal with bounded time intervals. We thus fix someT >0 and focus on D=D[0, T].

(Everything carries over toD[T1, T2]in an obvious fashion.) Throughout, · will denote the uniform norm. Two functionsg1, g2Dare close in theJ1-topologyif they are uniformly close after a small distortion of the domain.

Formally, letΛbe the set of increasing homeomorphismsλ:[0, T] → [0, T], and letλidΛdenote the identity. Then dJ1,T(g1, g2)=infλΛ{g1λg2λλid}defines a metric onDwhich induces theJ1-topology. While its restriction toC=C[0, T]coincides with the uniform topology, discontinuous functions areJ1-close to each other if they have jumps of similar size at similar positions.

In contrast, theM1-topologyallows a functiong1with a jump att to be approximated arbitrarily well by some continuousg2(with large slope neart). For convenience, we let[a, b]denote the (possibly degenerate) closed interval with endpoints a, b∈R, irrespective of their order. LetΓ (g):= {(t, x)∈ [0, T] ×R: x∈ [g(t), g(t)]}denote the completed graph ofg, and letΛ(g)be the set of all its parametrizations, that is, all continuousG=(λ, γ ):[0, T] → Γ (g) such thatt< t implies either λ(t) < λ(t )or λ(t)=λ(t ) plus |γ (t)g(λ(t))| ≤ |γ (t)g(λ(t))|. Then dM1,T(g1, g2)=infGi=ii)Λ(gi){λ1λ2γ1γ2}gives a metric inducingM1.

On the space D[0,) the τ-topology, τ ∈ {J1,M1}, is defined by the metric dτ,(g1, g2):=

0 et(1dτ,t(g1, g2))dt. Convergencegngin(D[0,), τ )means thatdτ,T(gn, g)→0 for every continuity pointT ofg.

For either topology, the corresponding Borelσ-fieldBD onD, generated by theτ-open sets, coincides with the usualσ-fieldBD generated by the canonical projections πt(g):=g(t). Therefore, any familyW =(Wt)t∈[0,T] or (Wt)t∈[0,) of real random variablesWt such that each pathtWt is cadlag, can be regarded as a random element ofD, equipped withτ =J1orM1.

2.2. Statement of the main result

Recall that for any ergodic measure preserving transformation (m.p.t.) f on a probability space (X, μ), and any YXwithμ(Y ) >0, thereturn timefunctionr:Y→N∪ {∞}given byr(y):=inf{k≥1:fk(y)Y}is integrable with mean

YrY =μ(Y )1(Kac’ formula), whereμY(A):=μ(YA)/μ(Y ). Moreover, thefirst return mapor induced mapF :=fr:YY is an ergodic m.p.t. on the probability space(Y, μY). This is widely used as a tool in the study of complicated systems, whereY is chosen in such a way thatF is more convenient thanf. In particular, given anobservable(i.e., a measurable function) φ:X→R, it may be easier to first consider itsinduced version Φ:Y →Ron Y, given by Φ:=r1

=0φf. By standard arguments, ifφL1(X, μ)thenΦL1(Y, μY)and


YΦY=μ(Y )1

Xφdμ. In this setup, we will denote the corresponding ergodic sums byφk:=k1

=0φfand Φn:=n1

j=0ΦFj, respectively.

Our core result allows us to pass from a weak invariance principle for the induced version to one for the original observable. Such a step requires some a priori control of the behaviour of ergodic sumsφkduring an excursion fromY. We shall express this in terms of the functionΦ:Y → [0,∞]given by

Φ(y):= max



∧ max


φ(y)φ(y) .

Note thatΦvanishes if and only if the ergodic sumsφkgrow monotonically (nonincreasing or nondecreasing) during each excursion. Hence boundingΦmeans limiting the growth ofφuntil the first return toY in at least one direction.

The expressionΦcan be understood also in terms of the maximal and minimal processesφ, φdefined during each excursion 0≤r(y)by

φ(y)= max


φ(y), φ(y)= min


Proposition 2.1.

(i) In the “predominantly increasing” case Φ(y) = max0r(y)(y)φ(y)), we have Φ(y) = max0r(y)(y)φ(y)).

(ii) In the “predominantly decreasing” case Φ(y) = max0r(y)(y)φ(y)), we have Φ(y) = max0r(y)(y)φ(y)).

Proof. This is immediate from the definition ofφandφ.

We useΦto impose a weak monotonicity condition forφduring excursions.

Theorem 2.2 (Inducing a weak invariance principle). Letf be an ergodic m.p.t.on the probability space(X, μ), and letYXbe a subset of positive measure with return timerand first return mapF.Suppose that the observable φ:X→Ris such that its induced versionΦsatisfies a WIP on(Y, μY)in that



Φt n B(n)

t0 LY)

W (t )

t0 in


, (2.1)

whereBis regularly varying of indexγ >0,and(W (t))t0is a process with cadlag paths.Moreover,assume that 1

B(n) max

0jnΦFj μ

Y 0. (2.2)

Thenφsatisfies a WIP on(X, μ)in that Wn(s)


φsn B(n)

s0 L(μ)


sμ(Y )

s0 in


. (2.3)

Remark 2.3 (α-stable processes). If the processW in(2.1)for the induced system is anα-stable Lévy process,then the limiting process in(2.3)is(


Remark 2.4. In general,the convergence from(2.3)fails in (D[0,),J1),even if (2.1)holds in theJ1-topology.

That this is the case for the intermittent maps(1.1)was pointed out in[33],Example2.1.

Remark 2.5 (Continuous sample paths). If the processW in(2.1) for the induced system has continuous sample paths,then the statement and proof of Theorem2.2is greatly simplified and the uniform topology(corresponding to the


uniform norm·)can be used throughout.In particular,the functionΦis replaced byΦ(y)=max0<r(y)|φ(y)|.

In the case of normal diffusionB(n)=n1/2,condition(2.2)is then satisfied ifΦL2. A simplified proof based on the one presented here is written out in[17],Appendix.

Remark 2.6 (Centering). In the applications that we principally have in mind(including the maps(1.1)),the observ- ableφ:X→Ris integrable,and hence so is its induced versionΦ:Y→R.In particular,ifφhas mean zero,thenΦ has mean zero and we are in a situation to apply Theorem2.2.From this,it follows easily that if condition(2.1)holds with

Pn(t)=Φt nt n


B(n) ,

and condition(2.2)holds withφreplaced throughout byφ

Xφin the definition ofΦ,then conclusion(2.3)is valid with



B(n) .

With a little more effort it is also possible to handle more general centering sequences where the process(Pn)in condition(2.1)takes the form

Pn(t)=Φt nt A(n) B(n) ,

for real sequencesA(n),B(n)withB(n)→ ∞.

The monotonicity condition (2.2) will be shown to hold, for example, if we have sufficiently good pointwise control for single excursions:

Proposition 2.7 (Pointwise weak monotonicity). Letf be an ergodic m.p.t.on the probability space(X, μ),and let YXbe a subset of positive measure with return timer.LetBbe regularly varying of indexγ >0.Suppose that for the observableφ:X→Rthere is someη(0,)such that for a.e.yY,

Φ(y)ηB r(y)

. (2.4)

Then the weak monotonicity condition(2.2)holds.

The proofs of Theorem2.2and Proposition2.7are given in Section3.

Assuming strong distributional convergenceLY)in (2.1), rather than⇒, is not a restriction, as an application ofμY the following result to the induced system(Y, μY, F )shows.

Proposition 2.8 (Automatic strong distributional convergence). Let f be an ergodic m.p.t.on aσ-finite space (X, μ).Letτ=J1orM1and letA(n),B(n)be real sequences withB(n)→ ∞.Assume thatφ:X→Ris measur- able,and that there is some probability measureP μand some random elementRofD[0,∞)such that


φt nt A(n) B(n)


P R in

D[0,), τ

. (2.5)

Then,Rn L(μ)

Rin(D[0,∞), τ ).

Proof. This is based on ideas in [14]. According to Zweimüller [38], Theorem 1, it suffices to check thatdτ,(Rnf, Rn)−→μ 0. The proof of [38], Corollary 3, shows that B(n)→ ∞ alone (that is, even without (2.5)) implies dJ1,(Rnf, Rn)−→μ 0. Since dM1,dJ1, (see [34], Theorem 12.3.2), the case τ =M1 then is a trivial



Remark 2.9. There is a systematic typographical error in[38]in that the factortin the centering processt A(n)/B(n) is missing,but the arguments there work,without any change,for the correct centering.

3. Proof of Theorem2.2

In this section, we give the proof of Theorem2.2and also Proposition2.7. Throughout, we assume the setting of Theorem2.2. In particular, we suppose thatf is an ergodic m.p.t. on the probability space(X, μ), and thatYXis a subset of positive measure with return timerand first return mapF.

3.1. Decomposing the processes

WhenY is chosen appropriately, many features off are reflected in the behaviour of the ergodic sumsrn=n1

j=0rFj, i.e., the times at which orbits return toY. These are intimately related to theoccupation timesorlap numbers




1Yf=max{n≥0: rnk} ≤k, k≥0.

The visits toY counted by theNkseparate the consecutiveexcursions fromY, that is, the intervals{rj, . . . , rj+1−1}, j≥0. Decomposing thef-orbit ofy into these excursions, we can represent the ergodic sums ofφas

φk=ΦNk+Rk onY with remainder termRk=k1

=rNkφf=φkrNkFNk encoding the contribution of the incomplete last excursion (if any). Next, decompose the rescaled processes accordingly, writing


withUk(s):=B(k)1ΦNsk, andVk(s):=B(k)1Rsk. On the time scale ofUn, the excursions correspond to the intervals[tn,j, tn,j+1),j ≥0, where tn,j:Y → [0,∞)is given bytn,j :=rj/n. Note that the interval containing a given pointt >0 is that withj=Nt n. Hence

t∈ [tn,Ntn, tn,Ntn+1) fort >0 andn≥1. (3.1) 3.2. Some almost sure results

In this subsection, we record some consequences of the ergodic theorem which we will use below. But first an ele- mentary observation, the proof of which we omit.

Lemma 3.1. Let(cn)n0be a sequence inRsuch thatn1cnc∈R.Define a sequence of functionsCn:[0,∞)→ Rby lettingCn(t):=n1ct nt c.Then,for anyT >0,(Cn)n1converges to0uniformly on[0, T].

For the occupation times ofY, we then obtain:

Lemma 3.2 (Strong law of large numbers for occupation times). The occupation timesNksatisfy (a) k1Nk−→μ(Y )a.e.onXask→ ∞.

(b) Moreover,for anyT >0, sup


k1Nt kt μ(Y )−→0 a.e.onXask→ ∞.

Proof. The first statement is immediate from the ergodic theorem. The second then follows by the preceding



Lemma 3.3. For anyT >0, limn→∞n1max0jT n+1(rFj)=0a.e.onY.

Proof. Applying the ergodic theorem toF and the integrable functionr, we getn1n1

j=0rFjμ(Y )1, and hence alson1(rFn)→0 a.e. onY. The result follows from Lemma3.1.

Pointwise control of monotonicity behaviour. We conclude this subsection by establishing Proposition2.7.

Proof of Proposition 2.7. We may suppose without loss that the sequence B(n) is nondecreasing. Since this se- quence is regularly varying,B(δn)/B(n)δγ for allδ >0. Hence forδ >0 fixed, there areδ >0 andn≥1 s.t.

ηB(h)/B(n) < δwhenevernnandhδn.

As a consequence of Lemma 3.3, there is somen≥1 such thatYn:= {n1max0jn(rFj) <δ}satisfies μY(Ync) < δfornn. In view of (2.4) we then see (using monotonicity ofBagain) that


B(n) max

0jnΦFj ≤ 1

B(n) max




B(n) < δ onYnfornn,

which proves (2.2).

3.3. Convergence of(Un)

As a first step towards Theorem 2.2, we prove that switching from Φt n to ΦNsk preserves convergence in the Skorohod space.

Lemma 3.4 (Convergence of(Un)). Under the assumptions of Theorem2.2, Un(s)

s0 μY


sμ(Y )

s0 in



Proof. Forn≥1 ands∈ [0,∞), we letun(s):=n1Nsn. Sinceun(s)n =Nsn, we have Un(s)=Pn


onY forn≥1 ands≥0. (3.2)

We regard Un, Pn, W, and un as random elements of (D,M1)=(D[0,),M1). Note that unD := {gD: g(0)≥0 andgnondecreasing}. Let u denote the constant random element of D given by u(s)(y):=sμ(Y ), s≥0.

Recalling Lemma3.2(b), we see that forμY-a.e.yY we haveun(·)(y)u(·)(y)uniformly on compact subsets of[0,∞). Hence,unuin(D,M1)holdsμY-a.e. In particular,

un μY

u in(D,M1).

By assumption (2.1) we also havePn LY)

Win(D,M1). But then we automatically get

(Pn, un)μY (W, u) in(D,M1)2, (3.3)

since the limituof the second component is deterministic.

The composition map(D,M1)×(D,M1)(D,M1),(g, v)gv, is continuous at every pair(g, v)with vC↑↑:= {gD: g(0)≥0 andgstrictly increasing and continuous}, cf. [34], Theorem 13.2.3. As the limit(W, u) in (3.3) satisfies Pr((W, u)∈D×C↑↑)=1, the standard mapping theorem for distributional convergence (cf. [34], Theorem 3.4.3) applies to(Pn, un), showing that

PnunμY Wu in(D,M1).

In view of (3.2), this is what was to be proved.


3.4. Control of excursions

Passing from convergence of(Un)to convergence of(Wn)requires a little preparation.

Lemma 3.5. (i)Letg, gD[0, T]and0=T0<· · ·< Tm=T.Then dM1,T

g, g

≤ max


g|[Tj1,Tj], g|[Tj1,Tj] .

(ii)LetgjD[Tj1, Tj]andg¯j:=1[Tj1,Tj)gj(Tj1)+1{Tj}gj(Tj).Then dM1,[Tj1,Tj](gj,g¯j)≤2gj+(TjTj1),


Proof. The first assertion is obvious. To validate the second, assume without loss thatj =1 and thatg1is predom- inantly increasing in thatg1=supT0stT1(g1(s)g1(t)). In this case, g1=supT0tT1(g1(t)g1(t)) for the nondecreasing functiong1(t):=supT0stg1(s). Therefore,

dM1,[T0,T1] g1, g1


Lettingg¯1:=1[T0,T1)g1(T0)+1{T1}g1(T1), it is clear that dM1,[T0,T1]

¯ g1,g¯1

g¯1− ¯g1=g

1(T1)g1(T1)g1. Finally, we check that

dM1,[T0,T1] g1,g¯1


To this end, we refer to Fig.1whereΓ1=Γ (g1)andΓ2=Γ (g¯1)represent the completed graphs ofg1andg¯1 respectively. HereΓ2consists of one horizontal line segment followed by one vertical segment. The picture ofΓ1is schematic, it may also contain horizontal and vertical line segments.

ChooseCon the graph ofΓ1that is equidistant fromADandDBand letEbe the point onDBthat is the same height asC. Choose parametrizationsGi=i, γi)ofΓi,i=1,2, satisfying

(i) G1(0)=G2(0)=A,G1(1)=G2(1)=B, (ii) G1(12)=C,G2(12)=E,

(iii) γ1(t)=γ2(t)for allt∈ [12,1].

Automaticallyλ1λ2 ≤ |AD| =T1T0and by constructionγ1γ2 ≤ |DE| ≤ |AD|, as required.

Fig. 1. A monotone excursion.


As a consequence, we obtain:

Lemma 3.6. dM1,T(Wn, Un)≤max0jT n+1(n1r+2B(n)1Φ)Fj.

Proof. LetyY and decompose[0, T]according to the consecutive excursions, lettingTj:=tn,j(y)T,jm:=

T n+1. Considerg(t):=Wn(t)(y),t∈ [0, T]. If we setgj:=g|[Tj1,Tj], theng¯jas defined in Lemma3.5coincides withUn(·)(y)|[Tj1,Tj], so that


Wn(·)(y), Un(·)(y)

≤ max


ButTjTj1=n1rFj, and sincegj(s)gj(t)=B(n)1φ)Fj(y), for suitable 0r, we see that Lemma3.5gives




as required.

Proof of Theorem2.2. Fix any T >0. By Lemma3.3,n1max0jT n+1(rFj)→0 a.e. and by assumption (2.2)B(n)1max0jT n+1Fj)μY 0. Hence Lemma3.6guarantees that

dM1,T(Wn, Un)μY 0. (3.4)

Recall also from Lemma3.4that Un(s)

0sT μY


sμ(Y )

0sT in

D[0, T],M1

. (3.5)

It follows (see [8], Theorem 3.1) from (3.4) and (3.5) that Wn(s)

0sT μY


sμ(Y )

0sT in

D[0, T],M1


This immediately gives (Wn(s))t0μY (W (sμ(Y )))s0 in(D[0,),M1). Strong distributional convergence as

asserted in (2.3) follows via Proposition2.8.

4. Examples

We continue to suppose thatf is an ergodic m.p.t. on a probability space(X, μ)with first return mapF =fr:YY whereμ(Y ) >0. Suppose further that the induced mapF:YY is Gibbs–Markov with ergodic invariant probability measureμY and partitionβ, and thatr|ais constant for eachaβ. Letφ:X→Rbe anLmean zero observable, with induced observableΦ:Y→R.

Theorem 4.1. Suppose thatφis constant onfa for everyaβ and∈ {0, . . . , r|a−1}.IfΦ lies in the domain of anα-stable law,thenΦ satisfies the WIP in(D,J1)withB(n)=n1/α.If in addition condition(2.2)holds,thenφ satisfies the WIP in(D,M1).

Proof. By [1],n1/αΦnconverges in distribution to the given stable law. The assumptions guarantee that the induced observable Φ is constant on eachYj. Hence we can apply Tyran-Kami´nska [33], Corollary 4.1, to deduce thatΦ satisfies the corresponding α-stable WIP in(D,J1). In particular, condition (2.1) is satisfied. The final statement

follows from Theorem2.2.

For certain examples, including Pomeau–Manneville intermittency maps, we can work with general Hölder observ- ables, thus improving upon [33], Example 4.1. The idea is to decompose the observableφinto a piecewise constant observableφ0and a Hölder observableφ˜in such a way that onlyφ0“sees” the source of the anomalous behaviour.


In the remainder of this section, we carry out this procedure for the maps (1.1) and thereby prove Theorem1.1.

(Lemma4.2and Proposition4.3below hold in the general context of induced Gibbs–Markov maps.)

Fix θ(0,1) and let dθ denote the symbolic metric on Y, so dθ(x, y)=θs(x,y) where s(x, y) is the least integer n≥0 such that Fnx, Fny lie in distinct elements of β. An observable Φ:Y →R is piecewise Lips- chitz if Da(Φ):=supx,ya,x=y|Φ(x)Φ(y)|/dθ(x, y) <∞ for each aβ, and Lipschitz if Φθ = |Φ|+ supaβDa(Φ) <∞. The space Lip of Lipschitz observablesΦ:Y→Ris a Banach space. Note thatΦ is integrable with

aβμY(a)Da(Φ) <∞if and only if

aβμY(a)1aΦθ<∞. LetLdenote the transfer operator forF:YY.

Lemma 4.2. (a)The essential spectral radius ofL: Lip→Lipis at mostθ. (b)Suppose thatΦ:Y→Ris a piecewise Lipschitz observable satisfying


Proof. This is standard. See, for example, [1], Theorem 1.6, for part (a) and [25], Lemma 2.2, for part (b).

Proposition 4.3. LetΦ:Y →R be a piecewise Lipschitz mean zero observable lying inLp,for somep(1,2).

Assume that

aβμY(a)1aΦθ<∞.Thenmaxj=0,...,n1nγΦjd0for allγ >1/p.

Proof. Suppose first thatF is weak mixing (this assumption is removed below). ThenL: Lip→Lip has no eigenval- ues on the unit circle except for the simple eigenvalue at 1 (corresponding to constant functions). By Lemma4.2(a), there existsτ <1 such that the remainder of the spectrum ofLlies strictly inside the ball of radiusτ. In particular, there is a constantC >0 such thatLnv

vYnvfor allv∈Lip,n≥1.

By Lemma4.2(b),∈Lip. Henceχ=

j=1LjΦ∈Lip. Following Gordin [16], writeΦ= ˆΦ+χFχ.

ThenΦˆ ∈Lp (sinceχ∈Lip andΦLp). ApplyingLto both sides and noting thatL(χF )=χ, we obtain that ˆ =0. It follows that the sequence{ ˆΦn;n≥1}defines a reverse martingale sequence.

By Burkholder’s inequality [9], Theorem 3.2,

| ˆΦn|p



Φˆ2Fj 1/2









| ˆΦ|pFj 1/p

= | ˆΦ|pn1/p.

By Doob’s inequality [13] (see also [9], Equation (1.4), p. 20),|maxj=0,...,n1Φˆj|p | ˆΦn|pn1/p. By Markov’s inequality, for >0 fixed,μY(|maxj=0,...,n1Φˆj| ≥nγ)≤ |maxj=0,...,n1Φˆj|pp/(pnγp)n(γp1)→0 asn

∞. Hence maxj=0,...,n1nγΦˆjd0. SinceΦandΦˆ differ by a bounded coboundary, maxj=0,...,n1nγΦjd0 as required.

It remains to remove the assumption about eigenvalues (other than 1) forLon the unit circle. Suppose that there are ksuch eigenvalues e,ω(0,),=1, . . . , k(including multiplicities). Then we can writeΦ=Ψ0+k


whereLnΨ0θnΨ0θand=eΨ. In particular, the above argument applies toΨ0, while=eΨ, =1, . . . , k.

A simple argument (see [24]) shows thatΨF =eΨ for=1, . . . , k, so that|n

j=1ΨFj|≤2|e− 1|1|Ψ|which is bounded inn. Hence the estimate forΦfollows from the one forΨ0. Proof of Theorem1.1. We verify the hypotheses of Theorem2.2. A convenient inducing set for the maps (1.1) isY = [12,1]. Letφ0=φ(0)μ(Y )1φ(0)1Y. (The first term is the important one, and the second term is simply an arbitrary choice that ensures thatφ0has mean zero while preserving the piecewise constant requirement in Theorem4.1.) Write φ=φ0+ ˜φand note thatφ˜ is a mean zero piecewise Hölder observable vanishing at 0. We have the corresponding decompositionΦ=Φ0+ ˜Φfor the induced observables. By Theorem4.1,Φ0satisfies the WIP (in theJ1topology).

Letηdenote the Hölder exponent ofφ. By the proof of [18], Theorem 1.3,φ˜induces to a piecewise Lipschitz mean zero observableΦ˜ satisfying

aβμY(a)1aΦ˜θ<∞for suitably chosenθ. Moreover [18] shows thatΦ˜ lies inL2




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