Nonconventional limit theorems in averaging

www.imstat.org/aihp 2014, Vol. 50, No. 1, 236–255

DOI:10.1214/12-AIHP514

© Association des Publications de l’Institut Henri Poincaré, 2014

Nonconventional limit theorems in averaging

Yuri Kifer

aInstitute of Mathematics, The Hebrew University, Jerusalem 91904, Israel. E-mail:kifer@math.huji.ac.il Received 9 September 2011; revised 4 June 2012; accepted 4 July 2012

Abstract. We consider “nonconventional” averaging setup in the form ^dX_dt^ε(t)=εB(X^ε(t), Ξ (q₁(t)), Ξ (q₂(t)), . . . , Ξ (q(t))) whereΞ (t), t≥0 is either a stochastic process or a dynamical system with sufficiently fast mixing whileq_j(t)=α_jt, α₁< α₂<

· · ·< α_k andq_j, j=k+1, . . . , grow faster than linearly. We show that the properly normalized error term in the “nonconven- tional” averaging principle is asymptotically Gaussian.

Résumé. Nous considérons un cadre non conventionnel de moyenne de la forme ^dX_dt^ε(t)=εB(X^ε(t), Ξ (q₁(t)), Ξ (q₂(t)), . . . , Ξ (q(t)))oùΞ (t), t≥0 est un processus stochastique ou un système dynamique suffisamment mélangeant tandis queq_j(t)= α_jt, α₁< α₂<· · ·< α_ketq_j, j=k+1, . . . , ont une croissance sur-linéaire. Nous montrons que le terme d’erreur après renor- malisation est asymptotiquement gaussien.

MSC:Primary 34C29; secondary 60F17; 37D20

Keywords:Averaging; Limit theorems; Martingales; Hyperbolic dynamical systems

1. Introduction

Nonconventional ergodic theorems (see [12]) known also after [2] as polynomial ergodic theorems studied the lim- its of expressions having the form 1/NN

n=1F^q1(n)f1· · ·F^q(n)fwhereF is a weakly mixing measure preserving transformation,f_i’s are bounded measurable functions andq_i’s are polynomials taking on integer values on the in- tegers. Originally, these results were motivated by applications to multiple recurrence for dynamical systems taking functionsf_i being indicators of some measurable sets and only convergence in theL²-sense was dealt with but later [1] provided also almost sure convergence under additional conditions. Recently such results were extended in [6] to the continuous time dynamical systems, i.e. to expressions of the form

1 T

_T

0

F^q1(t)f₁· · ·F^q(t)fdt,

whereF^sis now an ergodic measure preserving flow.

In this paper we consider the averaging setup X^ε(n+1)=X^ε(n)+εB

X^ε(n), Ξ q₁(n)

, . . . , Ξ q(n)

(1.1) in the discrete time case and

dX^ε(t) dt =εB

X^ε(t), Ξ q₁(t)

, . . . , Ξ q(t)

(1.2)

in the continuous time case with Ξ being either a stochastic process or having the formΞ (s)=F^sf whereF^s is a dynamical system andf is a function. Positive functionsq1, . . . , q will satisfy certain conditions which will be specified in the next section, in particular, firstkof them are linear while others grow faster than preceeding ones.

An example where (1.2) emerges is obtained when we consider a time dependent small perturbation of the oscillator equation

¨

x+λ²x=εg(x,x, t),˙ (1.3)

where the force termgdepends on time in a random wayg(x, y, t)=g(x, y, Ξ (q1(t)), . . . , Ξ (q(t))). Then passing to the polar coordinates(r, φ)withx=rsin(λ(t−φ))andx˙=λrcos(λ(t−φ))the equation (1.3) will be transformed into (1.2). It seems reasonable that a random force may depend on versions of a same process or a dynamical system moving with different speeds which is what we have here.

As it is well known (see, for instance, [29]), ifB(x, y1, . . . , y)is bounded and Lipschitz continuous inx and the limit

B(x)¯ = lim

T→∞

1 T

_T

0

B x, Ξ

q₁(t)

, . . . , Ξ q(t)

dt (1.4)

exists then for anyS≥0,

εlim→0 sup

0≤t≤S/ε

X^ε(t)− ¯X^ε(t)=lim

ε→0 sup

0≤t≤S

Z^ε(t)− ¯Z(t )=0, (1.5)

where dX¯^ε(t)

dt =εB¯X¯^ε(t)

and Z^ε(t)=X^ε(t /ε), Z(t )¯ = ¯X^ε(t /ε). (1.6)

In the discrete time case we have to take B(x)¯ = lim

N→∞

1 N

N n=0

B x, Ξ

q₁(n)

, . . . , Ξ q(n)

(1.7) and (1.5) remains true withX¯^ε given by (1.6) and (1.7). Almost everywhere limits in (1.4) and (1.7) follow from [22] under our (and even weaker) assumptions and in some relevant to our setup cases they could be derived by nonconventional pointwise ergodic theorems from [6] and [1].

After nonconventional ergodic theorems (or in the probabilistic language laws of large numbers) are established the next natural step is to obtain central limit theorem type results which was accomplished in [25]. The averaging prin- ciple (1.5) can be considered as an extension of the ergodic theorem since ifB(x, ξ₁, . . . , ξ)in (1.1) does not depend on x thenX^1/N(N )becomes the nonconventional average _N¹SN whereSN=

0≤n≤NB(Ξ (q1(n), . . . , Ξ (q(n)).

Now if _N¹SNconverges toB¯ asN→ ∞then convergence in distribution of√

N (X^1/N(N )− ¯B)to a normal random variable is, in fact, a nonconventional central limit theorem. The main goal of this paper is to extend the functional central limit theorem type results obtained in [25] for such sumsSNto the above nonconventional averaging setup in the spirit of what was done in the standard (conventional) averaging case in [18] and [20]. Central limit theorem type results turn in the averaging setup into assertions about Gaussian approximations of the slow motionX^εgiven by (1.1) or by (1.2) whereΞis a fast mixing stochastic process or a dynamical system while unlike the standard (conventional) case we have the processΞ taken simultaneously at different timesq_i(t)in the right hand side of (1.1) and (1.2).

We prove, first, our limit theorems for stochastic processes under rather general conditions resembling the definition of mixingales (see [27] and [28]) and then check these conditions for more familiar classes of stochastic processes and dynamical systems. In [25] we imposed mixing assumptions in a standard way relying on two parameter families of σ-algebras (see [5]) while our assumptions here use only filtrations (i.e. nondecreasing families) ofσ-algebras which are easier to construct for various classes of dynamical systems. As one of applications we check some form of our conditions for Anosov flows which serve as fast motions in our nonconventional averaging setup where we rely on the notion of Markov families from [8] and [9].

At the end of the paper we discuss a fully coupled averaging setup in our nonconventional situation where already an averaging principle itself becomes a problem.

2. Preliminaries and main results

Our setup consists of a ℘-dimensional stochastic process {Ξ (t ), t ≥0 ort =0,1, . . .} on a probability space (Ω,F, P r) together with a filtration ofσ-algebrasFl⊂F,0≤l≤ ∞so thatFl⊂Fl ifl≤l. For convenience we extend the definitions ofFl given only forl≥0 to negativel by definingFl=F0 forl <0. In order to relax required stationarity assumptions to some kind of weak “limiting stationarity” our setup includes another probability measureP on the space(Ω,F). Namely, we assume that the distribution ofΞ (t )with respect toP does not depend ontand the joint distribution of{Ξ (t ), Ξ (t)}fort≥tdepends only ont−twhich can be written in the form

Ξ (t )P=μ and

Ξ (t ), Ξ t

P =μ_t−t for allt≥t, (2.1)

whereμis a probability measure onR^℘andμ_s, s≥0 is a probability measure onR^℘×R^℘.

Our setup relies on two probability measuresP r andP in order to include, for instance, Markov processesΞ (t ) satisfying the Doeblin condition (see [16] or [10]) starting at a fixed point or with another noninvariant distribution.

ThenP rwill be a corresponding probability in the path space whileP will be the stationary probability constructed by the initial distribution being the invariant measure ofΞ (t ). Usual mixing conditions for stochastic processes are formulated in terms of a double parameter family ofσ-algebras via a dependence coefficient between widely separated past and futureσ-algebras (cf. [5] and [25]) but this approach often is not convenient for applications to dynamical systems where natural futureσ-algebras do not seem to exist unless an appropriate symbolic representation is avail- able. By this reason we formulate below a different set of mixing and approximation conditions for the processΞ which seem to be new and will enable us to treat some of dynamical systems models within a class of stochastic processes satisfying our assumptions.

In order to avoid some of technicalities we restrict ourselves here mostly to bounded functions though our results can be obtained for more general classes of functions with polynomial growth supplemented by appropriate moment boundedness conditions similarly to [25]. For any functiong=g(ξ,ξ )˜ onR^℘×R^℘introduce its Hölder norm

|g|κ=supg(ξ,ξ )˜ + |g(ξ,ξ )˜ −g(ξ,ξ˜)|

|ξ−ξ|^κ+ |˜ξ− ˜ξ|^κ: ξ =ξ, ξ =ξ . (2.2) Here and in what follows |ψ − ˜ψ|^κ for two vectors ψ=(ψ1, . . . , ψ) and ψ˜ =(ψ˜1, . . . ,ψ˜) denotes the sum

i=1|ψ_i− ˜ψ_i|^κ. Next, forp, q≥1 ands≥0 we define a sort of a mixing coefficient η_p,κ,s(n)=sup

t≥0

E g

Ξ (n+t ), Ξ (n+t+s)

|F_[t]

−E_Pg

Ξ (n+t ), Ξ (n+t+s)

p: g=g(ξ,ξ ),˜ |g|κ≤1

, η_p,κ(n)=η_p,κ,0(n), (2.3) where · p is the L^p-norm on the space (Ω,F, P r),[·] denotes the integral part and throughout this paper we writeE for the expectation with respect toP r andE_P for the expectation with respect toP. We will need also an (one-sided) approximation coefficient

ζ_q(n)=sup

t≥0

E

Ξ (t )|F_[t]+n

−Ξ (t )

q. (2.4)

Assumption 2.1. Givenκ∈(0,1]there existp, q≥1andm, δ >0satisfying γ_m=EΞ (0)^m<∞, 1

2≥ 1 p+ 2

m+δ

q, δ < κ−

p, κq >1 (2.5)

with=(−1)℘and such that ∞

n=0

n

η¹p,κ^{−/(pθ )}(n)+ζ_q^δ(n)

<∞ and lim

n→∞ηp,κ,s(n)=0 for alls≥0, (2.6)

where_p < θ < κ.

Next, letB=B(x, ξ )=(B⁽¹⁾(x, ξ ), . . . , B^(d)(x, ξ )), ξ=(ξ₁, . . . , ξ)∈R^℘be ad-vector function onR^d×R^℘ such that for some constantK >0 and allx,x˜∈R^d,ξ,ξ˜∈R^℘,i, j, l=1, . . . , d,

B⁽ⁱ⁾(x, ξ )≤K, B⁽ⁱ⁾(x, ξ )−B⁽ⁱ⁾(x,˜ ξ )˜ ≤K

|x− ˜x| + j=1

|ξ_j− ˜ξ_j|^κ

and

∂B⁽ⁱ⁾(x, ξ )

∂xj

≤K,

∂²B⁽ⁱ⁾(x, ξ )

∂xj∂xl

≤K. (2.7)

We will be interested in the central limit theorem type results asε→0 for the solutionX^ε(t)=X^ε_x(t)of the equation dX^ε(t)

dt =εB

X^ε(t), Ξ q1(t)

, Ξ q2(t)

, . . . , Ξ q(t)

, X^ε_x(0)=x, t∈ [0,T/ε], (2.8) whereq₁(t) < q₂(t) <· · ·< q(t), t >0 are increasing functions such thatq_j(t)=α_jt forj ≤k < withα₁< α₂<

· · ·< α_k whereas the remainingq_js grow faster int. Namely, we assume similarly to [25] that for anyγ >0 and k+1≤i≤,

tlim→∞

q_i(t+γ )−q_i(t)

= ∞ (2.9)

and

tlim→∞

q_i(γ t)−q_i−1(t)

= ∞. (2.10)

Set B(x)¯ =

B(x, ξ1, . . . , ξ)dμ(ξ1)· · ·dμ(ξ). (2.11)

We consider also the solutionX¯^ε(t)= ¯X^ε_x(t)of the averaged equation dX¯^ε(t)

dt =εB¯X¯^ε(t)

, X¯_x^ε(0)=x. (2.12)

It will be convenient to denoteZ^ε(t)=X^ε(t /ε),Z(t )¯ = ¯X^ε(t /ε)and to introduceY^ε(t)=Y_y^ε(t)by Y_y^ε(t)=y+

t 0

BZ(s), Ξ¯

q1(s/ε) , Ξ

q2(s/ε)

, . . . , Ξ

q(s/ε)

ds. (2.13)

Theorem 2.2. Suppose that (2.7), (2.9), (2.10) and Assumption 2.1 hold true. Then the family of processes G^ε(t)=ε^−1/2(Y_z^ε(t)− ¯Z_z(t)), t ∈ [0,T] converges weakly as ε →0 to a Gaussian process G⁰(t), t ∈ [0,T] having not necessarily independent increments (see an example in [25]) with covariances of its components G⁰(t)=(G^0,1(t ), . . . , G^0,d(t)) having the form EG^0,l(s)G^0,m(t))=_{min(s,t )}

0 A^l,m(u)du with the matrix function {A^l,m(u),1≤l, m≤d}computed in Section4.Furthermore,the family of processesQ^ε(t)=ε^−1/2(Z^ε(t)− ¯Z(t)), t∈ [0,T]converges weakly asε→0to a Gaussian processQ⁰(t), t∈ [0,T]which solves the equation

Q⁰(t)=G⁰(t)+ t

0

∇ ¯BZ(s)¯

Q⁰(s)ds. (2.14)

In the discrete time setup(1.1)the similar results hold true assuming thatq_i’s take on integer values on integers, γ in(2.9)is replaced by1,α_i is replaced byifori=1, . . . , kand definingZ^ε(t)=X^ε([t /ε])together withY^ε=Y_y^ε given by

Y_y^ε(t)=y+ _t

0

BZ(s), Ξ¯ q₁

[s/ε] , Ξ

q₂ [s/ε]

, . . . , Ξ q

[s/ε]

ds (2.15)

while leaving all other definitions and assumptions the same as above.

Observe that we work withB¯ defined by (2.11) but in our circumstances the law of large numbers from [22] yields B¯ also as an almost sure limit in (1.4) and (1.7) even under weaker conditions than here. Note also that we need the full strength of (2.6) only for one argument in Section 4borrowed from [18] but for a standard limit theorem not in the averaging setup, i.e. whenB(x, ξ₁, . . . , ξ)=B(ξ₁, . . . , ξ)does not depend onx, it suffices to require only summability of the expression in brackets in (2.6).

An important point in the proof of the first part of Theorem2.2is to introduce the representation

B(x, ξ )= ¯B(x)+B₁(x, ξ₁)+ · · · +B(x, ξ₁, . . . , ξ), (2.16) whereξ=(ξ₁, . . . , ξ)and fori < ,

Bi(x, ξ1, . . . , ξ)

=

B(x, ξ₁, . . . , ξ)dμ(ξi+1)· · ·dμ(ξ)−

B(x, ξ₁, . . . , ξ)dμ(ξi)· · ·dμ(ξ) (2.17) while

B(x, ξ₁, . . . , ξ)=B(x, ξ₁, . . . , ξ)−

B(x, ξ₁, . . . , ξ)dμ(ξ). (2.18)

Next, set Y_i^ε(t)=

_{t /αi}

0

Bi

Z(s), Ξ¯

q1(s/ε) , Ξ

q2(s/ε)

, . . . , Ξ

q(s/ε)

ds fori=1, . . . , k

(2.19) while fori=k+1, . . . , set Y_i^ε(t)=

_t

0

B_iZ(s), Ξ¯

q₁(s/ε) , Ξ

q₂(s/ε)

, . . . , Ξ

q(s/ε) ds withY₀^ε(t)=Y_0,y^ε (t)=y+t

0Bi(Z(s))¯ ds. ThusY_y^εfrom (2.13) has the representation Y_y^ε(t)=Y₀^ε(t)+

k i=1

Y_i^ε(α_it )+ i=k+1

Y_i^ε(t). (2.20)

We consider also X^ε₀(t)= ¯X^ε(t), X_i^ε(t)=X_i,x^ε (t)=x +ε_t

0B_i(X^ε_i(s), Ξ (q₁(s)), . . . , Ξ (q(s)))ds and Z^ε_i(t)= X_i^ε(t /ε)for alli≥0. Fori≥1 set also

G^ε_i(t)=ε^−1/2Y_i^ε(t) and Q^ε_i(t)=ε^−1/2Z_i^ε(t). (2.21) Relying on martingale approximations (which also can be done employing mixingales from [27] and [28]) we will show that any linear combinationk

i=1λiG^ε_i converges weakly asε→0 to a Gaussian processk

i=1λiG⁰_i. It turns out that in the continuous time case eachG^ε_i, i=k+1, . . . , converges weakly asε→0 to zero, and so the processes Y_i^ε, i > k do not play any role in the limit. It follows thatG^ε converges weakly to a Gaussian process G⁰ such thatG(t )=_k

i=1λ_iG⁰_i(α_it ). On the other hand, in the discrete time case eachG^ε_i, i > k cannot be disregarded, in general, and it converges weakly asε→0 to a Gaussian processG⁰_i which is independent of any otherG⁰_j. The above difference between discrete and continuous time cases is due to the different natural forms of the assumption (2.9) in these two cases. These arguments yield the first part of Theorem2.2while its second part concerning convergence of Q^ε asε→0 is proved via some Taylor expansion and approximation arguments.

In order to clarify the role of the coefficientsη_p,κ andζ_q we compare them with the more familiar mixing and approximation coefficients defined via a two parameter family ofσ-algebrasGs,t∈F,−∞ ≤s≤t≤ ∞by

_p(n)= sup

s≥0,g

E(g|G−∞,s)−E_Pg

p: gisGs+n,∞-measurable and|g| ≤1

(2.22)

and

βq(n)=sup

t≥0

E

Ξ (t )|Gt−n,t+n

−Ξ (t )

q, (2.23)

respectively, whereGst⊂Gst ifs≤sandt≥t. Then settingFl=G_−∞,l we obtain by the contraction property of conditional expectations that

βq(n)≥sup

t≥0

E

Ξ (t )|Gt−n,t+n

−Ξ (t )+Ξ (t )−E

Ξ (t )|G_−∞,[t]+n+1

q

≥ζq(n+1)−βq(n) i.e. βq(n)≥1

2ζq(n+1). (2.24)

Furthermore, g

Ξ (n+t ), Ξ (n+t+s)

−g E

Ξ (n+t )|Gn+t−[n/2],n+t+[n/2] , E

Ξ (n+t+s)|Gn+t+s−[n/2],n+t+s+[n/2]

p≤2|g|κβ_pκ^κ [n/2]

,

and so

η_p,κ(n)≤ _p

[n/2]

+2β_pκ^κ [n/2]

|g|κ. (2.25)

Thus, appropriate conditions on decay of coefficientspandβqas in [25] yield corresponding conditions onηp,κand ζ_q. The other direction does not hold true but still it turns out that most of the technique from [25] can be employed in our circumstances, as well.

The conditions of Theorem2.2hold true for many important stochastic processes. In the continuous time case they are satisfied when, for instance,Ξ (t )=f (Υ (t ))whereΥ (t)is either an irreducible continuous time finite state Markov chain or a nondegenerate diffusion process on a compact manifold whilef is a Hölder continuous vector function. In the discrete time case we can take, for instance, Ξ (n)=f (Υ (n)) withΥ (n) being a Markov chain satisfying the Doeblin condition (see, for instance, [16], pp. 367–368). In all these examplesη_p,κ(n)andζ_q(n)decay innexponentially fast while (2.6) requires much less. In fact, in both casesΞ (t )may depend on whole paths of a Markov processΥ assuming only certain weak dependence on their tails.

Important classes of processes satisfying our conditions come from dynamical systems. In Section 6 we take Ξ (t )=Ξ (t, z)=g(F^tz) whereF^t is a C² Anosov flow (see [24]) on a compact manifold M whose stable and unstable foliations are jointly nonintegrable andgis a Hölder continuous℘-vector function onM. It turns out that if we take the initial pointzon an elementS of a Markov family (see Section6) introduced in [8] distributed there at random according to a probability measure equivalent to the volume onS then Assumption2.1can be verified.

This does not yield though a desirable limit theorem where the initial point is taken at random on the whole manifold M distributed according to the Sinai–Ruelle–Bowen (SRB) measure (or the normalized Riemannian volume). We observe that a suspension representation of Anosov flows employed in [20] to derive limit theorems in the conventional averaging setup does not work in our situation becauseF^qi(t)x, i=1, . . . , arrive at the ceiling of the suspension at different times for differenti’s.

In the discrete time case there are several important classes of dynamical systems where our conditions can be verified. First, for transformations where symbolic representations via Markov partitions are available (Axiom A diffeomorphisms (see [3]) and expanding endomorphisms, some one-dimensional maps e.g. the Gauss map (see [15]) etc.) we can rely on standard mixing and approximation assumptions based on two parameter families ofσ-algebras as in (2.22) and (2.23). On the other hand, for many transformations Markov partitions are not available but still it is possible to construct one parameter increasing or decreasing filtration ofσ-algebras so that our conditions can be verified. For some classes of noninvertible transformationsF it is possible to choose an appropriate initialσ-algebra F0such thatF⁻¹F0⊂F0and then to define a decreasing filtrationFi=F⁻ⁱF0(see [26] and [13]). Passing to the natural extension as in Remark 3.12 of [13] we can turn to an increasing filtration and to verify our conditions. On the other hand, our results can be derived under appropriate conditions with respect to decreasing families ofσ-algebras.

Namely, letF⊃F0⊃F1⊃F2⊃ · · ·and define mixing and approximation coefficients by η_p,κ,s(n)=sup

t≥s

E g

Ξ (t ), Ξ (t−s)

|F_[t]+n

−EPg

Ξ (t ), Ξ (t−s)

p: g=g(ξ,ξ ),˜ |g|κ≤1

, ηp,κ(n)=ηp,κ,0(n) (2.26)

and

ζ_q(n)=sup

t≥n

E

Ξ (t )|F[t]−n

−Ξ (t )

q. (2.27)

Then under Assumption2.1we can rely on estimates of Section3below and in place of martingales there arrive at reverse martingales and employ a limit theorem for the latter.

Remark 2.3. IfB¯ ≡0then according to Theorem2.2the processX^ε(t)is very close to its initial point on the time interval of order1/ε.Thus,in order to see fluctuations of order1it makes sense to consider longer time and to deal withV^ε(t)=X^ε(t /ε²).Under the stronger condition

B(x, ξ₁, . . . , ξ)dμ(ξ)≡0 it is not difficult to mimic the proofs in[19]and [4]relying on the technique of Sections3and 4below in order to obtain thatV^ε(t), t∈ [0, T] converges weakly asε→0to a diffusion process with parameters obtained in the same way as in[19]and[4].It is not clear whether,in general,this result still holds true assuming only thatB¯ ≡0.Though most of the required estimates still go through in the latter case a convergence ofV^ε to a Markov process seems to be problematic in a general nonconventional averaging setup.

3. Estimates and martingale approximation

The proof of Theorem2.2will employ a modification of the machinery developed in [25]. First, we have to study the asymptotical behavior asε→0 of

G^ε_i(t)=√ ε

_τi(t)/ε

0

Bi

Z(εs), Ξ¯ q1(s)

, . . . , Ξ qi(s)

ds (3.1)

which is obtained from the definition (2.21) by the change of variabless→s/εand whereτ_i(t)=t /α_ifori=1, . . . , k andτ_i(t)=t fori=k+1, . . . , . Observe that if _N¹₊₁≤ε≤_N¹ andN≥1 then by (2.7),

G^ε_i(t)−G^1/N_i (t)≤2Kt d

√N (3.2)

and so it suffices to study the asymptotical behavior ofG^1/N_i asN→ ∞. Set I_i,N(n)=

_n+1

n

B_iZ(s/N ), Ξ¯ q₁(s)

, . . . , Ξ q_i(s)

ds. (3.3)

In view of (2.7) the asymptotical behavior ofG^1/N_i asN→ ∞is the same as ofN^−1/2S_i,N(t)where S_i,N(t)=

[N τ_i(t)] n=0

I_i,N(n). (3.4)

There are two obstructions for applying directly the results of [25] to the sum (3.4). First, unlike [25] the integrand in (3.3) depends on the “slow time”s/N. Secondly, our mixing and approximation coefficients look differently from the corresponding coefficients in [25]. Still, it turns out that these obstructions can be dealt with and after minor modifications the method of [25] start working in our situation, as well. Namely, the dependence on the “slow time”

being deterministic will not prevent us from making estimates similar to [25] while dependence of I_i^N on N will just require us to deal with martingale arrays which creates no problems as long as we obtain appropriate limits of

variances and covariances. Concerning the second obstruction we observe that one half of the approximation estimate from [25] is contained in the coefficientζpwhile another half is hidden in the coefficientηp,κwhich also suffices for required mixing estimates.

We explain next more precisely why estimates similar to [25] hold true in our circumstances, as well. Letf (ψ, ξ,ξ )˜ be a function onR×R^℘×R^℘such that for anyψ, ψ∈Randξ,ξ , ξ˜ ,ξ˜∈R^℘,

f (ψ, ξ,ξ )˜ −f

ψ, ξ,ξ˜≤Cψ−ψ^κ+ξ−ξ^κ+y−y^κ

and f (ψ, ξ,ξ )˜ ≤C. (3.5) Then settingg(ψ )=E_Pf (ψ, Ξ (0), Ξ (s))we obtain from (2.1) and (2.3) that for allu, v≥0 andn∈N,

E f

ψ, Ξ (n+u), Ξ (n+u+v)

|F[u]

−g(ψ )

p≤Cη_p,κ,v(n). (3.6)

Leth(ψ, ω)=E(f (ψ, Ξ (n+u), Ξ (n+u+v))|F_[u])−g(ψ ). Then by (3.5) we can choose a version ofh(ψ, ω) such that with probability one simultaneously for allψ, ψ∈R,

h(ψ, ω)−h

ψ, ω≤2Cψ−ψ^κ. (3.7)

Since, in addition,h(ψ, ω)p≤Cη_p,κ(n)by (3.6) for allψ∈R, we obtain by Theorem 3.4 from [25] that for any random-vectorΨ=Ψ (ω),

hΨ (ω), ω

a≤cC

ηp,κ,v(n)1−/(pθ )

1+ Ψm

, (3.8)

where _p< θ < κ, ¹_a≥_p¹+_m¹ andc=c(, p, κ, θ ) >0 depends only on parameters in brackets. Since hΨ (ω), ω˜

=E

fΨ , Ξ (n˜ +u), Ξ (n+u+v)

|F_[u]

(ω) a.s. (3.9)

providedΨ˜ isF[u]-measurable we obtain from (3.6)–(3.9) together with the Hölder inequality (cf. Corollary 3.6(ii) in [25]) that,

E f

Ψ, Ξ (n+u), Ξ (n+u+v)

|F[u]

−g(Ψ )

a

≤C

η_p,κ,v(n)1−/(pθ )

1+ Ψm

+2CΨ −E(Ψ|F_[u])^δ

q (3.10)

provided ¹_a≥_p¹+_m² +^δ_q.

We apply the above estimates in two cases. First, when f (ψ, ξ,ξ )˜ =f (ψ, ξ )=B_i(x, ξ₁, . . . , ξ_i) with ψ = (ξ1, . . . , ξi−1)∈R^(i−1)℘,ξ =ξi∈R^℘,n= [(qi(t)−qi−1(t))/2],u=qi(t)−nandΨ =(Ξ (q1(t)), Ξ (q2(t)), . . . , Ξ (q_i−1(t)). In the second casef (ψ, ξ,ξ )˜ =B_i(x, ξ₁, . . . , ξ_i)B_j(y, ξ₁, . . . , ξ_j)withψ=(ξ₁, . . . , ξ_i−1, ξ₁, . . . , ξ_j−1)

∈R^(i+j−2)℘,ξ=ξi,ξ˜=ξ_j∈R^℘,n= [(min(qi(t), qj(s))−max(qi−1(t), qj−1(s)))/2]whenn >0,u=min(qi(t), q_j(s))−nandΨ =(Ξ (q₁(t)), . . . , Ξ (q_i−1(t)), Ξ (q₁(s)), . . . , Ξ (q_j−1(s))). The estimates for the first case are used for martingale approximations while the second case emerges when computing covariances.

Since

B_i(x, ξ₁, . . . , ξ_i−1, ξ_i)dμ(ξi)=0 we obtain by (3.10) the estimate E

B_i Ξ

q₁(t)

, . . . , Ξ q_i(t)

|F[q_i(t)]−n

a≤C

η_p,κ(n)1−(pθ )

+

ζ_q(n)δ

(3.11) for someC >0 independent oft wheren=ni(t)= [(qi(t)−qi−1(t))/2]. Next, for anyx∈R,ξ1, . . . , ξi−1∈R andr=1,2, . . .set

Bi,r

x, ξ1, . . . , ξi−1, Ξ (t )

=E Bi

x, ξ1, . . . , ξi−1, Ξ (t )

|F_[t]+r

and Ξr(t)=E

Ξ (t )|F_[t]+r

. Then by (2.4) and (2.7) together with the Hölder inequality,

B_i

x, ξ₁, . . . , ξ_i−1, Ξ (t )

−B_i,r

x, ξ₁, . . . , ξ_i−1, Ξ (t )

q

≤2Bi

x, ξ1, . . . , ξi−1, Ξ (t )

−Bi

x, ξ1, . . . , ξi−1, E

Ξ (t )|F_[t]+r

q

≤2KdΞ (t )−E Ξ (t )

|F_[t]+r)^κ

q≤2Kdζ_q^δ(r). (3.12)

This together with the last part of Theorem 3.4 in [25] yields that B_i

x, Ξ q₁(t)

, . . . , Ξ q_i(t)

−B_i,r x, Ξ_r

q₁(t)

, . . . , Ξ_r

q_i−1(t)

a≤cζ_q^δ(r) (3.13)

provided ¹_a ≥ ¹_p+_m² +^δ_q andδ <min(κ,1− _pκ^d )wherec=c(δ, a, p, q) >0 depends only on the parameters in brackets. Set

b^l,m_ij (x, y;s, t)=E B_i^(l)

x, Ξ q1(s)

, . . . , Ξ qi(s)

B_j^(m) y, Ξ

q1(t)

, . . . , Ξ qj(t)

,

where, recall,B_i^(l)is thel-th component of thed-vectorB_i. Now, by (2.7), (3.11) and (3.13), b^l,m_ij (x, y;s, t)≤C

η_p,κ(n)1−/(pθ )

+

ζ_q(n)δ

, (3.14)

whereC >0 does not depend ons, t≥0 andn=n_ij(s, t)=max(nˆ_ij(s, t),nˆ_{j i}(t, s))withnˆ_ij(s, t)= [¹₂min(qi(s)− q_j(t), q_i(s)−q_i−1(s))].

Now, set I_i,N,r(n)=

_n

n−1

B_i,rZ(s/N ), Ξ¯ _r q₁(s)

, . . . , Ξ_r

q_i−1(s) ds,

S_i,N,r(t)=

[N τ_i(t)] n=1

I_i,N,r(n),

(3.15) R_i,r(m)= ^∞

l=m+1

E

I_i,N,r(l)|Fm+r

,

D_i,N,r(m)=I_i,N,r(m)+R_i,r(m)−R_i,r(m−1) and M_i,N,r(t)=

[N τi(t)] n=1

D_i,N,r(n).

In view of (2.6) and (3.11) applied with a=2 we see that the series for R_i,r(m) converges inL², D_i,N,r(m)is Fm+r-measurable and sinceE(Di,N,r(m)|Fm−1+r)=0 we obtain that{Di,N,r(m),Fm+r}0≤m≤[N τ_i(T )]is a martingale differences array. Next, we rely on (3.11) and the inequality

S_i,N,r(t)−M_i,N,r(t)≤R_i,r

N τ_i(t)+R_i,r(0)

to observe that asN→ ∞the limiting behavior ofN^−1/2Si,N,r asN→ ∞is the same as ofN^−1/2Mi,N,r. Once we derive that appropriate covariances converge asN→ ∞, which will be done in the next section, we can invoke for the latter expression a version of the functional central limit theorem for martingale arrays (see, for instance, Section 2 in Ch. VIII of [17]) yielding thatN^−1/2M_i,N,r, and so also N^−1/2S_i,N,r converge asN → ∞to ad-dimensional Gaussian process with independent increments. Next, we write

Si,N(t)=Si,N,1(t)+ ∞ r=1

Si,N,2^r(t)−S_i,N,2r−1(t)

(3.16) and relying on uniform moment estimates of the terms in this series similar to Proposition 5.9 of [25] we obtain that also N^−1/2Si,N converges in distribution as N → ∞to a d-dimensional Gaussian process with independent increments. Next, by a version of the Cramér–Wold argument (see Corollary 5.7 in [25]) we obtain that(S_i,N, i= 1, . . . , ) converges in distribution asN → ∞to a d-dimensional Gaussian process with independent increments while covariances computations in the next section show that last−krandomd-vectors of the limiting process are independent of each other and of other randomd-vectors there. Thus, we will end up with limitingd-dimensional

Gaussian processesG⁰_i, i=1, . . . , with independent increments such that(G⁰_i, i=1, . . . , )is and-dimensional Gaussian process andG⁰_k+1, . . . , G⁰are independent of each other and ofG⁰_i withi≤k. Now we write

G⁰(t)= k i=1

G⁰_i(it)+ i=k+1

G⁰_i(t)= k j=1

k i=1

λ_ij

G⁰_j(it)−G⁰_j

(i−1)t ,

whereλ_ij=1 ifi≤j andλ_ij=0, otherwise, obtaining from the above thatG⁰is a Gaussiand-dimensional process (for more details see [25]).

Observe that under a bit stronger assumptions we could employ in our setup another martingale approximation construction from [23] which is based on increasing blocks and negligible gaps between them and which does not re- quirer-approximations as above. In order to complete this programm it remains only to compute limiting covariances as in Section 4 of [25] taking care also of the slow times/N entering (3.3) and (3.15).

4. Limiting covariances

In this section we show the existence and compute the limit asε→0 of the expression E

G^ε_i,l(s)G^ε_j,m(t)

=ε τ_i(s/ε)

0

τ_j(t /ε) 0

b^l,m_ij Z(εu),¯ Z(εv)¯ ;u, v

dudv. (4.1)

We start with showing that there exists a constantC >0 such that for allt ≥s >0, l=1, . . . , d,N ≥1 and i= 1, . . . , ,

sup

N≥1

EG^1/N

i,l (t)−G^1/N_i,l (s)²≤C(t−s). (4.2)

In order to obtain (4.2) we note that by (2.9) and (2.10) fort≥s,

q_i(t)−q_i(s)≥α_i(t−s) and q_i(t)−q_i−1(t)≥α_i−1t wheni=2, . . . , k (4.3) and for anyγ >0 there existstγ such that for allt≥tγ andi=k+1, . . . , ,

q_i(t)−q_i(s)≥(t−s)+γ⁻¹ and q_i(t)−q_i−1(t)≥t+γ⁻¹. (4.4) Now (4.2) follows from (2.6), (3.14), (4.1), (4.3) and (4.4). Observe, that by (3.2) and (4.1) if_N¹₊₁≤ε≤_N¹ then

EG^ε_i,l(s)G^ε_j,m(t)−EG^1/N_i,l (s)G^1/N_j,m(t)≤4KdC√

√ T

N ,

and so it suffices to study (4.1) asε=_N¹ andN→ ∞.

Next, we claim that ifi > jandi > kthen the limit in (4.1) as ¹_ε=N→ ∞exists and equals zero. Indeed, in this case for any smallγ >0 withγ T ≤s,

EG^1/N_i,l (s)G^1/N_j,m(t)≤I₁+I₂, (4.5)

where by (4.2), I₁=EG^1/N

i,l (γT)G^1/N_j,m(t)≤ E

G^1/N_i,l (γT)21/2 E

G^1/N_j,m(t)21/2

≤C

γTt (4.6)

and by (3.14), I₂=E

G^1/N_i,l (s)−G^1/N_i,l (γT)

G^1/N_j,m(t)

= 1 N

_sN

γTN

du _{τj(tN )}

0

b^l,m_ij Z(u/N ),¯ Z(v/N )¯ ;u, v dv≤ C

N _sN

γTN

du _{τj(tN )}

0

ρ_ij(u, v)dv, (4.7)