Central and non-central limit theorems for weighted power variations of fractional Brownian motion

(1)

arXiv:0710.5639v3 [math.PR] 22 Aug 2009

fra tional Brownian motion IvanNourdin

∗

,David Nualart

†‡

and Ciprian A.Tudor

§

Abstra t: In this paper, we prove some entral and non- entral limit theorems for renormalized weighted power variations of order

q ≥ 2

of the fra tional Brownian motion with Hurst parameter

H ∈ (0, 1)

, where

q

is an integer. The entral limit holds for

1 2q

< H ≤ 1 −

2q

1

, the limit being a onditionally Gaussian distribution. If

H <

1 2q

weshowthe onvergen e in

L

2

to alimit whi h only depends on the fra tional Brownian motion, and if

H > 1 −

1 2q

we show the onvergen e in

L

2

to a sto hasti integralwithrespe ttotheHermitepro essoforder

q

.

Key words: fra tional Brownian motion, entral limittheorem, non- entral limit theorem, Hermite pro ess.

2000Mathemati sSubje t Classi ation: 60F05,60H05,60G15,60H07. Thisversion: August2009.

1 Introdu tion

The study of single path behaviorof sto hasti pro esses is often basedon thestudy of their powervariations,andthereexistsaveryextensiveliteratureonthesubje t. Re allthat,areal

q > 0

beinggiven,the

q

-powervariationofasto hasti pro ess

X

,withrespe ttoasubdivision

π

n

= {0 = t

n,0

< t

n,1

< . . . < t

n,κ(n)

= 1}

of

[0, 1]

,isdened to bethesum

κ(n)

_X

k=1

|X

t

n,k

− X

t

n,k−1

|

q

_.

For simpli ity, onsider from now on the ase where

t

n,k

= k2

−n

for

n ∈ {1, 2, 3, . . .}

and

k ∈ {0, . . . , 2

n

_}

. In thepresent paperwe wish to point out some interesting phenomena when

X = B

is a fra tional Brownian motion of Hurst index

H ∈ (0, 1)

, and when

q ≥ 2

is an

integer. In fa t, we will also drop the absolute value (when

q

is odd) and we will introdu e

∗

LaboratoiredeProbabilitésetModèlesAléatoires, UniversitéPierre etMarieCurie,Boîte ourrier188, 4 Pla eJussieu,75252 ParisCedex5,Fran e,ivan.nourdinupm .fr

†

Department of Mathemati s,University of Kansas, 405 Snow Hall, Lawren e, Kansas 66045-2142, USA, nualartmath.ku.edu

‡

TheworkofD.NualartissupportedbytheNSFGrantDMS-0604207

§

(2)

2 n

X

k=1

f (B

_(k−1)2

−n

)(∆B

_k2

−n

)

q

,

q ∈ {2, 3, 4, . . .},

(1.1)

where the fun tion

f : R → R

is assumed to be smooth enough and where

∆B

k2

−n

denotes, hereandinall the paper,thein rement

B

k2

−n

− B

_(k−1)2

−n

.

The analysis of the asymptoti behavior of quantities of type (1.1 ) is motivated, for instan e, by the study of the exa t rates of onvergen e of some approximation s hemes of s alar sto hasti dierential equations driven by

B

(see [7℄, [12 ℄ and [13 ℄) besides, of ourse, thetraditional appli ationsof quadrati variations to parameter estimation problems.

Now,letusre allsomeknownresults on erning

q

-powervariations(for

q = 2, 3, 4, . . .

), whi h aretodaymoreor less lassi al. First,assumethat theHurst indexis

H =

1

2

,thatis

B

isastandardBrownianmotion. Let

µ

q

denotethe

q

thmomentofastandardGaussianrandom variable

G ∼ N (0, 1)

. By the s aling property of the Brownian motionand using the entral limittheorem, itis immediatethat, as

n → ∞

:

2 −n/2

2 n

X

k=1

h

(2

n/2

∆B

_k2

−n

)

q

− µ

_q

i

_Law

−→ N (0, µ

2q

− µ

2 q

).

(1.2)

Whenweightsareintrodu ed,aninterestingphenomenonappears: insteadofGaussianrandom variables, we rather obtain mixing random variables aslimit in (1.2 ). Indeed, when

q

iseven

and

f : R → R

is ontinuous andhaspolynomialgrowth, itisa veryparti ular ase ofa more

general resultbyJa od [10℄ (seealso Se tion 2inNourdin andPe ati[16 ℄ forrelatedresults) thatwe have, as

n → ∞

:

2 −n/2

2 n

X

k=1

f (B

_(k−1)2

−n

)

h

(2

n/2

∆B

_k2

−n

)

q

− µ

_q

i

_Law

−→

q

µ

2q

− µ

2 q

Z

1

0 f (B

s

)dW

s

.

(1.3)

Here,

W

denotes another standard Brownian motion, independent of

B

. When

q

is odd, still

for

f : R → R

ontinuous withpolynomial growth,we have,this time,as

n → ∞

:

2 −n/2

2 n

X

k=1

f (B

_(k−1)2

−n

)(2

n/2

∆B

_k2

−n

)

q Law

−→

Z

1

0 f (B

s

)

q

µ

2q

− µ

2 q+1

dW

s

+ µ

q+1

dB

s

,

(1.4)

seefor instan e[16 ℄.

Se ondly,assumethat

H 6=

1

2

,thatisthe asewherethefra tional Brownianmotion

B

hasnot independent in rementsanymore. Then(1.2 ) hasbeenextendedbyBreuerand Major [1℄,Dobrushin and Major [5℄,Giraitis and Surgailis [6℄ or Taqqu [21 ℄. Pre isely,ve ases are onsidered, a ordingto the evenness of

q

and thevalue of

H

:

(3)

•

if

q

iseven and if

H =

3

4

,as

n → ∞

,

1 √

n

2 −n/2

2 n

X

k=1

(2

3

4 n

∆B

_k2

−n

)

q

− µ

_q

−→ N (0, e

Law

σ

2 ₃

4 ,q

).

(1.6)

•

if

q

iseven and if

H ∈ (

3

4 , 1)

, as

n → ∞

,

2 n−2nH

2 n

X

k=1

(2

nH

∆B

k2

−n

)

q

− µ

_q

Law

−→

Hermiter.v.. (1.7)

•

if

q

isoddand if

H ∈ (0,

1

2 ]

,as

n → ∞

,

2 −n/2

2 n

X

k=1

(2

nH

∆B

_k2

−n

)

q Law

−→ N (0, e

σ

2 _H,q

).

(1.8)

•

if

q

isoddand if

H ∈ (

1

2 , 1)

, as

n → ∞

,

2 −nH

2 n

X

k=1

(2

nH

∆B

_k2

−n

)

q Law

−→ N (0, e

σ

2 _H,q

).

(1.9)

Here,

e

σ

H,q

> 0

denotesome onstant depending only on

H

and

q

. The term Hermite r.v. denotes a randomvariablewhose distributionisthesame asthatof

Z

(2)

at timeone,for

Z

(2)

dened inDenition 7 below.

Now, let us pro eed with the results on erning the weighted power variations in the asewhere

H 6=

1

2

. Consider the following ondition on a fun tion

f : R → R

,where

q ≥ 2

is aninteger:

(H

q

) f

belongs to

C

2q

and, for any

p ∈ (0, ∞)

and

0 ≤ i ≤ 2q

:

sup

t∈[0,1]

E

|f

(i)

(B

t

)|

p

< ∞

.

Suppose that

f

satises

(H

q

)

. If

q

is even and

H ∈ (

1

2 ,

3

4 )

, then by Theorem 2 in León and Ludeña [11℄ (see also Cor uera et al [4℄ for related results on theasymptoti behavior of the

p

-variationof sto hasti integrals withrespe t to

B

)wehave,as

n → ∞

:

2 −n/2

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

nH

∆B

_k2

−n

)

q

− µ

_q

Law

−→ e

σ

H,q

Z

1

0 f (B

s

)dW

s

,

(1.10)

(4)

thisdire tionhasbeenobservedbyGradinaruetal [9℄. Namely,if

q ≥ 3

isoddand

H ∈ (0,

1

2 )

, wehave,as

n → ∞

:

2 nH−n

2 n

X

k=1

f (B

_(k−1)2

−n

)(2

nH

∆B

_k2

−n

)

q

L

2 −→ −

µ

q+1

₂

Z

1

0 f

′

(B

s

)ds.

(1.11)

Also,when

q = 2

and

H ∈ (0,

1

4 )

,Nourdin [14 ℄ proved thatwe have, as

n → ∞

:

2 2Hn−n

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

nH

∆B

_k2

−n

)

2 − 1

L

2 −→

1 ₄

Z

1

0 f

′′

(B

s

)ds.

(1.12)

In view of (1.3 ), (1.4 ), (1.10 ), (1.11 ) and (1.12 ), we observe that the asymptoti be-haviors of the power variations of fra tional Brownian motion (1.1 ) an be really dierent, depending onthe valuesof

q

and

H

. The aimof thepresent paperisto investigate what hap-pensinthewholegeneralitywithrespe tto

q

and

H

. Ourmain toolistheMalliavin al ulus thatappeared, inseveralre ent papers, to be very useful inthestudy of thepower variations for sto hasti pro esses. As we will see, the Hermite polynomials play a ru ial role in this analysis. In the sequel, for an integer

q ≥ 2

, we write

H

q

for the Hermite polynomial with degree

q

dened by

H

q

(x) =

(−1)

q

q!

e

x2

2 d

q

dx

q

e

−

x2

2 ,

andwe onsider,when

f : R → R

isadeterministi fun tion,thesequen eofweighted Hermite variationof order

q

dened by

V

_n

(q)

(f ) :=

2 n

X

k=1

f B

_(k−1)2

−n

H

q

2 nH

∆B

k2

−n

.

(1.13)

Thefollowing isthe main resultofthis paper.

Theorem 1 Fix an integer

q ≥ 2

, andsuppose that

f

satises

(H

q

)

. 1. Assume that

0 < H <

1 2q

. Then, as

n → ∞

, it holds

2 nqH−n

V

_n

(q)

(f )

_−→

L

2 (−1)

q

2 q

_q!

Z

1

0 f

(q)

(B

s

)ds.

(1.14) 2. Assume that

1 2q

< H < 1 −

2q

1

. Then, as

n → ∞

, it holds

B, 2

−n/2

V

_n

(q)

(f )

_{−→ B, σ}

Law

H,q

Z

1

0 f (B

s

)dW

s

,

(1.15)

where

W

isa standard Brownianmotion independent of

B

and

(5)

3. Assume that

H = 1 −

1 2q

. Then, as

n → ∞

, it holds

B,

√

1 n

2 −n/2

_V

(q)

n

(f )

Law

−→ B, σ

1−1/(2q),q

Z

1

0 f (B

s

)dW

s

,

(1.17)

where

W

isa standard Brownianmotion independent of

B

and

σ

_1−1/(2q),q

=

2 log 2

q!

1 −

1 2q

q

1 −

1 _q

q

.

H > 1 −

1 2q

. Then, as

n → ∞

, it holds

2 nq(1−H)−n

V

_n

(q)

(f )

_−→

L

2 Z

1

0 f (B

s

)dZ

s

(q)

,

(1.19) where

Z

(q)

denotes the Hermite pro ess of order

q

introdu ed in Denition7 below. Remark 1. When

q = 1

, we have

V

(1)

n

(f ) = 2

−nH

P

2 n

k=1

f B

(k−1)2

−n

∆B

_k2

−n

. For

H =

1

2

,

2 nH

_V

(1)

n

(f )

onverges in

L

2

to the It sto hasti integral

R

1

0 f (B

s

)dB

s

. For

H >

1

2

,

2 nH

V

n

(1)

(f )

onverges in

L

2

andalmost surelyto theYoung integral

R

1

0 f (B

s

)dB

s

. For

H <

1

2

,

2 3nH−n

V

n

(1)

(f )

onverges in

L

2

to

−

1

2 R

1

0 f

′

(B

s

)ds

. Remark 2. In the riti al ase

H =

1 2q

(

q ≥ 2

), we onje ture the following asymptoti

behavior: as

n → ∞

,

B, 2

−n/2

V

_n

(q)

(f )

_{−→ B, σ}

Law

1/(2q),q

Z

1

0 f (B

s

)dW

s

+

(−1)

q

2 q

_q!

Z

1

0 f

(q)

(B

s

)ds

,

(1.20)

for

W

a standard Brownian motion independent of

B

and

σ

1/(2q),q

the onstant dened by (1.16 ). A tually,(1.20 )for

q = 2

and

H =

1

4

hasbeen proved in[2, 15 ,17℄ afterthat therst draftofthe urrentpaperhavebeensubmitted. Thereaderisalsoreferredto [16 ℄for thestudy ofthe weighted variationsasso iated withiterated Brownian motion, whi h is anon-Gaussian self-similarpro ess of order

1

4

. When

H

is between

1

4

and

3

4

,one an renepoint 2 ofTheorem 1asfollows:

Proposition 2 Let

q ≥ 2

be an integer,

f : R → R

be a fun tion su h that

(H

q

)

holds and assumethat

H ∈ (

1

4 ,

3

4 )

. Then

B, 2

−n/2

V

_n

(2)

(f ), . . . , 2

−n/2

V

_n

(q)

(f )

(1.21)

Law

−→

B, σ

H,2

Z

1

0 f (B

s

)dW

s

(2)

, . . . , σ

H,q

Z

1

0 f (B

s

)dW

s

(q)

,

where

(W

(2)

_{, . . . , W}

(q)

₎

is a

(q − 1)

-dimensional standard Brownian motion independent of

B

(6)

understandingof the asymptoti behaviorofweightedpower variations offra tional Brownian motion:

Corollary 3 Let

q ≥ 2

be aninteger,and

f : R → R

beafun tionsu hthat

(H

q

)

holds. Then,

as

n → ∞

: 1. When

H >

1

2

and

q

isodd,

2 −nH

2 n

X

k=1

f (B

_(k−1)2

−n

)(2

nH

∆B

_k2

−n

)

q L

2 −→ qµ

q−1

Z

1

0 f (B

s

)dB

s

= qµ

q−1

Z

B

1

0 f (x)dx.

(1.22) 2. When

H <

1

4

and

q

iseven,

2 2nH−n

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

nH

∆B

_k2

−n

)

q

− µ

_q

L

2 −→

1 ₄

q

2 µ

_q−2

Z

1

0 f

′′

(B

s

)ds.

(1.23)

(We re over (1.12) by hoosing

q = 2

). 3. When

H =

1

4

and

q

iseven,

B, 2

−n/2

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

n/4

∆B

_k2

−n

)

q

− µ

_q

!

Law

−→

B,

1

4 q

2 µ

_q−2

Z

1

0 f

′′

(B

s

)ds

+e

σ

_1/4,q

Z

1

0 f (B

s

)dW

s

,

(1.24)

where

W

isastandardBrownianmotionindependentof

B

and

e

σ

1/4,q

isthe onstantgiven by (1.26) justbelow. 4. When

1

4 < H <

3

4

and

q

iseven,

B, 2

−n/2

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

nH

∆B

_k2

−n

)

q

− µ

_q

!

Law

−→

B, e

σ

H,q

Z

1

0 f (B

s

)dW

s

,

(1.25) for

W

a standard Brownian motionindependentof

B

and

(7)

5. When

H =

3

4

and

q

iseven,

B,

_√

1 n

2 −n/2

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

nH

∆B

_k2

−n

)

q

− µ

_q

!

Law

−→

B, e

σ

3

4 ,q

Z

1

0 f (B

s

)dW

s

,

(1.27) for

W

a standard Brownian motionindependentof

B

and

e

σ

3

4 ,q

=

v

u

t

q

X

p=2

2 log 2 p!

q

p

2 µ

2 q−p

1 −

1 2q

q

1 −

1 q

q

.

6. When

H >

3

4

and

q

iseven,

2 n−2Hn

2 n

X

k=1

f (B

_(k−1)2

−n

)

(2

nH

∆B

_k2

−n

)

q

− µ

_q

L

2 −→ 2µ

q−2

q

2 Z

1

0 f (B

s

)dZ

s

(2)

,

(1.28) for

Z

(2)

the Hermite pro ess introdu ed in Denition 7.

Finally, we an also give a new proof of the following result, stated and proved by Gradinaruet al. [8℄and Cheriditoand Nualart[3℄in a ontinuous setting:

Theorem 4 Assume that

H >

1

6

, and that

f : R → R

veries (

H

6

). Then the limit in

probability, as

n → ∞

, of the symmetri Riemannsums

1

2

2 n

X

k=1

f

′

(B

_k2

−n

) + f

′

(B

_(k−1)2

−n

)

∆B

_k2

−n

(1.29)

existsand isgiven by

f (B

1 ) − f(0)

. Remark 3 When

H ≤

1

6

, quantity (1.29) does not onverge in probability in general. As a ounterexample, one an onsider the ase where

f (x) = x

3

, see Gradinaru et al. [8 ℄ or CheriditoandNualart [3 ℄.

2 Preliminaries and notation

Webrieyre allsomebasi fa tsaboutsto hasti al uluswithrespe ttoafra tionalBrownian motion. One refers to [19 ℄ for further details. Let

B = (B

t

)

t∈[0,1]

be a fra tional Brownian motionwithHurst parameter

H ∈ (0, 1)

. That is,

B

is azero meanGaussian pro ess,dened ona ompleteprobabilityspa e

(Ω, A, P )

,withthe ovarian e fun tion

R

H

(t, s) = E(B

t

B

s

) =

1

2 s

2H

_{+ t}

2H

(8)

We suppose that

A

is the sigma-eld generated by

B

. Let

E

be the set of step fun tions on

[0, T ]

,and

H

bethe Hilbertspa edened asthe losureof

E

withrespe tto theinner produ t

h1

[0,t]

, 1

[0,s]

i

H

= R

H

(t, s).

The mapping

1 [0,t]

7→ B

t

an be extendedto an isometry between

H

and theGaussian spa e

H

1

asso iatedwith

B

. Wewill denote thisisometry by

ϕ 7→ B(ϕ)

.

Let

S

be the setof allsmooth ylindri al randomvariables,i.e. ofthe form

F = φ(B

t

1 , . . . , B

t

m

)

where

m ≥ 1

,

φ : R

m

_{→ R ∈ C}

∞

b

and

0 ≤ t

1 < . . . < t

m

≤ 1

. Thederivative of

F

withrespe t

to

B

istheelement of

L

2 _{(Ω, H)}

dened by

D

s

F =

m

X

i=1

∂φ

∂x

i

(B

t

1 , . . . , B

t

m

)1

[0,t

i

]

(s),

s ∈ [0, 1].

Inparti ular

D

s

B

t

= 1

[0,t]

(s)

. For any integer

k ≥ 1

, we denoteby

D

k,2

the losure of theset ofsmooth randomvariables withrespe tto thenorm

kF k

2 k,2

= E(F

2 ) +

k

X

j=1

E

_kD

j

_{F k}

2 _H

⊗j

.

TheMalliavin derivative

D

satisesthe hain rule. If

ϕ : R

n

_{→ R}

is

C

1 b

and if

(F

i

)

i=1,...,n

isa sequen eofelements of

D

1,2

,then

ϕ(F

1 , . . . , F

n

) ∈ D

1,2

and we have

D ϕ(F

1 , . . . , F

n

) =

n

X

i=1

∂ϕ

∂x

i

(F

1 , . . . , F

n

)DF

i

.

Wealsohavethefollowingformula,whi h aneasilybeprovedbyindu tionon

q

. Let

ϕ, ψ ∈ C

q

b

(

q ≥ 1

),and x

0 ≤ u < v ≤ 1

and

0 ≤ s < t ≤ 1

. Then

ϕ(B

t

− B

s

)ψ(B

v

− B

u

) ∈ D

q,2

and

D

q

ϕ(B

t

− B

s

)ψ(B

v

− B

u

)

=

q

X

a=0

q

a

ϕ

(a)

(B

t

− B

s

)ψ

(q−a)

(B

v

− B

u

)1

⊗a

_[s,t]

⊗1

e

⊗(q−a)

_[u,v]

,

(2.30)

where

⊗

e

meansthe symmetri tensor produ t.

The divergen e operator

I

is the adjoint of the derivative operator

D

. If a random variable

u ∈ L

2 _{(Ω, H)}

belongs to thedomain of thedivergen e operator,that is, ifitsatises

|EhDF, ui

H

| ≤ c

u

p

E(F

2 ₎

for any

F ∈ S ,

then

I(u)

isdened bythe duality relationship

E F I(u)

_{= E hDF, ui}

H

(9)

forevery

F ∈ D

1,2

.

For every

n ≥ 1

, let

H

n

be the

n

th Wiener haos of

B,

that is, the losed linear subspa e of

L

2 _{(Ω, A, P )}

generated by the random variables

{H

n

(B (h)) , h ∈ H, khk

H

= 1}

, where

H

n

is the

n

th Hermite polynomial. The mapping

I

n

(h

⊗n

) = n!H

n

(B (h))

provides a linearisometrybetween thesymmetri tensor produ t

H

⊙n

(equipped withthemodiednorm

k · k

H

⊙n

=

√

1 n!

k · k

H

⊗n

) and

H

n

. For

H =

1

2

,

I

n

oin ideswiththemultipleWiener-Itintegral oforder

n

. Thefollowing dualityformulaholds

E (F I

n

(h)) = E hD

n

F, hi

H

⊗n

,

(2.31)

foranyelement

h ∈ H

⊙n

and any randomvariable

F ∈ D

n,2

.

Let

{e

k

, k ≥ 1}

bea ompleteorthonormal systemin

H

. Given

f ∈ H

⊙n

and

g ∈ H

⊙m

,

forevery

r = 0, . . . , n ∧ m

,the ontra tion of

f

and

g

oforder

r

is theelement of

H

⊗(n+m−2r)

dened by

f ⊗

r

g =

∞

X

k

1 ,...,k

r

=1

hf, e

k

1 ⊗ . . . ⊗ e

k

r

i

H

⊗r

⊗ hg, e

_k

₁

⊗ . . . ⊗ e

_k

_r

i

_H

⊗r

.

Noti e that

f ⊗

r

g

is not ne essarily symmetri : we denote its symmetrization by

f e

⊗

r

g ∈

H

⊙(n+m−2r)

. We have the following produ tformula: if

f ∈ H

⊙n

and

g ∈ H

⊙m

then

I

n

(f )I

m

(g) =

n∧m

_X

r=0

r!

n

r

m

r

I

_n+m−2r

(f e

_⊗

r

g).

(2.32)

Were all the following simpleformulafor any

s < t

and

u < v

:

E ((B

t

− B

s

)(B

v

− B

u

)) =

1

2 |t − v|

2H

+ |s − u|

2H

− |t − u|

2H

− |s − v|

2H

.

(2.33)

Wewill also need the following lemmas:

Lemma 5 1. Let

s < t

belong to

[0, 1]

. Then,if

H < 1/2

, one has

E B

u

(B

t

− B

s

)

≤ (t − s)

2H

(2.34) for all

u ∈ [0, 1]

. 2. For all

H ∈ (0, 1)

,

2 n

X

k,l=1

E B

_(k−1)2

−n

∆B

_l2

−n

= O(2

n

).

(2.35)

3. For any

r ≥ 1

,we have, if

H < 1 −

(10)

4. For any

r ≥ 1

,we have, if

H = 1 −

1 2r

,

2 n

X

k,l=1

|E (∆B

k2

−n

∆B

_l2

−n

)|

r

= O(n2

2n−2rn

).

(2.37)

Proof: Toprove inequality (2.34),we justwrite

E(B

u

(B

t

− B

s

)) =

1

2 (t

2H

− s

2H

) +

1

2 |s − u|

2H

− |t − u|

2H

,

andobservethatwe have

|b

2H

_{− a}

2H

_{| ≤ |b − a|}

2H

for any

a, b ∈ [0, 1]

,be ause

H <

1

2

. Toshow (2.35 )using(2.33 ), we write

2 n

X

k,l=1

E B

_(k−1)2

−n

∆B

_l2

−n

= 2

−2Hn−1

2 n

X

k,l=1

|l − 1|

2H

− l

2H

− |l − k + 1|

2H

+ |l − k|

2H

≤ C2

n

,

thelast bound oming from a teles oping sum argument. Finally, to show (2.36 ) and (2.37), wewrite

2 n

X

k,l=1

|E (∆B

k2

−n

∆B

_l2

−n

)|

r

= 2

−2nrH−r

2 n

X

k,l=1

|k − l + 1|

2H

+ |k − l − 1|

2H

− 2|k − l|

2H

r

≤ 2

n−2nrH−r

∞

X

p=−∞

|p + 1|

2H

+ |p − 1|

2H

− 2|p|

2H

r

,

and observe that, sin e the fun tion

|p + 1|

2H

_{+ |p − 1|}

2H

_{− 2|p|}

2H

behavesas

C

H

p

2H−2

for large

p

,theseries intheright-handside is onvergent be ause

H < 1 −

1 2r

. Inthe riti al ase

H = 1 −

2r

1

,this series isdivergent,and

2 n

X

p=−2

n

|p + 1|

2H

_{+ |p − 1|}

2H

_{− 2|p|}

2H

r

behavesasa onstant time

n

. Lemma 6 Assume that

H >

1

2

.

1. Let

s < t

belong to

[0, 1]

. Then

E B

u

(B

t

− B

s

)

≤ 2H(t − s)

(2.38)

(11)

2. Assume that

H > 1 −

1 2l

for some

l ≥ 1

. Let

u < v

and

s < t

belong to

[0, 1]

. Then

|E(B

u

− B

v

)(B

t

− B

s

)| ≤ H(2H − 1)

2 2Hl + 1 − 2l

1 l

(u − v)

l−1

l

(t − s).

H > 1 −

1 2l

for some

l ≥ 1

. Then

2 n

X

i,j=1

E ∆B

i2

−n

∆B

_j2

−n

l

= O(2

2n−2ln

).

(2.40) Proof: We have

E B

u

(B

t

− B

s

)

=

1

2 t

2H

− s

2H

+

1

2 |s − u|

2H

− |t − u|

2H

.

But,when

0 ≤ a < b ≤ 1

:

b

2H

_{− a}

2H

= 2H

Z

_b−a

0 (u + a)

2H−1

_{du ≤ 2H b}

2H−1

_{(b − a) ≤ 2H(b − a).}

Thus,

|b

2H

_{− a}

2H

_{| ≤ 2H|b − a|}

and therstpoint follows.

Con erningthe se ond point, using Hölderinequality,we an write

|E(B

u

− B

v

)(B

t

− B

s

)| = H(2H − 1)

Z

v

u

Z

t

s

|y − x|

2H−2

_dydx

≤ H(2H − 1)|u − v|

l−1

l

Z

1

0 Z

t

s

|y − x|

2H−2

_dy

l

dx

!

1 l

≤ H(2H − 1)|u − v|

l−1

l

|t − s|

l−1

l

Z

1

0 Z

t

s

|y − x|

(2H−2)l

_dydx

1 l

.

Denoteby

H

′

_{= 1 + (H − 1)l}

andobserve that

H

′

_>

1

2

(be ause

H > 1 −

1 2l

). Sin e

2H

′

_{− 2 =}

(2H − 2)l

,we anwrite

H

′

(2H

′

_{− 1)}

Z

1

0 Z

t

s

|y − x|

(2H−2)l

_{dydx = E}

_B

H

′

1 (B

H

′

t

− B

H

′

s

)

≤ 2H

′

|t − s|

bytherst point of thislemma. Thisgivesthedesired bound. Weprove nowthethird point. We have

(12)

andthefun tion

|k +1|

2H

_{+ |k −1|}

2H

_−2|k|

2H

behavesas

|k|

2H−2

forlarge

k

. Asa onsequen e, sin e

H > 1 −

1 2l

,thesum

2 n

₋₁

X

k=−2

n

₊₁

|k + 1|

2H

+ |k − 1|

2H

− 2|k|

2H

l

behavesas

2 (2H−2)ln+n

andthethird pointfollows.

Now, let us introdu e the Hermite pro ess of order

q ≥ 2

appearing in (1.19 ). Fix

H > 1/2

and

t ∈ [0, 1]

. The sequen e

ϕ

n

(t)

n≥1

,dened as

ϕ

n

(t) = 2

nq−n

1 q!

[2

n

_t]

X

j=1

1 ⊗q

_[(j−1)2

−n

_,j2

−n

_]

,

isaCau hy sequen einthe spa e

H

⊗q

. Indeed,sin e

H > 1/2

,wehave

h1

[a,b]

, 1

[u,v]

i

H

= E (B

b

− B

a

)(B

v

− B

u

)

= H(2H − 1)

Z

b

a

Z

v

u

|s − s

′

_|

2H−2

_dsds

′

_,

sothat,for any

m ≥ n

(13)

Denition 7 Fix

q ≥ 2

and

H > 1/2

. The Hermite pro ess

Z

(q)

_{= (Z}

(q)

t

)

t∈[0,1]

of order

q

is dened by

Z

(q)

t

= I

q

(µ

(q)

t

)

for

t ∈ [0, 1]

. Let

Z

(q)

n

be thepro ess dened by

Z

(q)

n

(t) = I

q

(ϕ

n

(t))

for

t ∈ [0, 1]

. By onstru tion,it is lear that

Z

(q)

n

(t)

L

2 −→ Z

(q)

_(t)

as

n → ∞

,forallxed

t ∈ [0, 1]

. Ontheotherhand,itfollows,

from Taqqu [21 ℄ and Dobrushin and Major [5℄, that

Z

(q)

n

onverges in law to the standard and histori al

q

th Hermite pro ess, dened through its moving average representation as a multiple integral with respe t to a Wiener pro ess with time horizon

R

. In parti ular, the pro essintrodu edinDenition7hasthesamenitedimensionaldistributionsasthehistori al Hermitepro ess.

Letusnallymentionthatit anbeeasilyseenthat

Z

(q)

is

q(H −1)+1

self-similar,has

stationary in rements and admits moments of all orders. Moreover, it has Hölder ontinuous pathsoforder stri tly lessthan

q(H − 1) + 1

. For further results,werefer to Tudor [22 ℄.

3 Proof of the main results

Inthisse tionwewill providethe proofsofthemain results. Fornotational onvenien e,from now on, we write

ε

(k−1)2

−n

(resp.

δ

k2

−n

) insteadof

1 [0,(k−1)2

−n

_]

(resp.

1 [(k−1)2

−n

_,k2

−n

_]

). The following proposition provides information on the asymptoti behaviorof

E

V

n

(q)

(f )

2

,as

n

tendsto innity, for

H ≤ 1 −

1 2q

.

Proposition 8 Fix an integer

q ≥ 2

. Suppose that

f

satises (

H

q

). Then,if

H ≤

1 2q

, then

E

V

_n

(q)

(f )

2 = O(2

n(−2Hq+2)

).

(3.41) If

1 2q

≤ H < 1 −

2q

1

, then

E

V

_n

(q)

(f )

2 = O(2

n

).

(3.42) Finally,if

H = 1 −

1 2q

, then

E

V

_n

(q)

(f )

2 = O(n2

n

).

(3.43)

Proof. Using the relation between Hermite polynomials and multiple sto hasti integrals, we have

H

q

2 nH

_∆B

k2

−n

=

_q!

1

2 qnH

I

q

δ

_k2

⊗q

−n

. Inthis waywe obtain

(14)

tionship(2.31 )between themultiplesto hasti integral

I

N

andtheiteratedderivativeoperator

D

N

,obtaining

E

V

_n

(q)

(f )

2 =

2 2Hqn

q!

2

2 n

X

k,l=1

q

X

r=0

r!

q

r

2 ×E

n

f (B

_(k−1)2

−n

) f (B

_(l−1)2

−n

) I

_2q−2r

δ

⊗q−r

_k2

−n

⊗δ

e

_l2

⊗q−r

−n

o

hδ

k2

−n

, δ

_l2

−n

i

r

_H

= 2

2Hqn

2 n

X

k,l=1

q

X

r=0

1 r!(q − r)!

2 ×E

nD

D

2q−2r

f (B

_(k−1)2

−n

) f (B

_(l−1)2

−n

)

, δ

_k2

⊗q−r

−n

⊗δ

e

⊗q−r

_l2

−n

E

H

⊗(2q−2r)

o

hδ

k2

−n

, δ

_l2

−n

i

r

_H

,

where

⊗

e

denotes the symmetrization of the tensor produ t. By (2.30 ), the derivative of the produ t

D

2q−2r

_{f (B}

(k−1)2

−n

) f (B

_(l−1)2

−n

)

isequal to a sumof derivatives:

D

2q−2r

f (B

_(k−1)2

−n

) f (B

_(l−1)2

−n

)

=

X

a+b=2q−2r

f

(a)

(B

_(k−1)2

−n

) f

(b)

(B

_(l−1)2

−n

)

×

(2q − 2r)!

a!b!

ε

⊗a

(k−1)2

−n

⊗ε

e

⊗b

_(l−1)2

−n

.

Wemake thede omposition

(15)

D

n

= 2

2Hqn

q

X

r=1

X

a+b=2q−2r

2 n

X

k,l=1

E

n

f

(a)

(B

_(k−1)2

−n

) f

(b)

(B

_(l−1)2

−n

)

o

_{(2q − 2r)!}

r!(q − r)!

2 _a!b!

×hε

⊗a

_(k−1)2

−n

⊗ε

e

⊗b

_(l−1)2

−n

, δ

_k2

⊗q−r

−n

⊗δ

e

⊗q−r

_l2

−n

i

H

⊗(2q−2r)

hδ

_k2

−n

, δ

_l2

−n

i

r

_H

,

for some ombinatorial onstants

α(c, d, e, f )

. That is,

A

n

and

B

n

ontain all thetermswith

r = 0

and

(a, b) = (q, q)

;

C

n

ontains thetermswith

r = 0

and

(a, b) 6= (q, q)

;and

D

n

ontains

theremaining terms.

For anyinteger

r ≥ 1

,we set

α

n

=

sup

k,l=1,...,2

n

hε

(k−1)2

−n

, δ

_l2

−n

i

_H

,

(3.45)

β

r,n

=

2 n

X

k,l=1

hδ

k2

−n

, δ

_l2

−n

i

H

r

,

(3.46)

γ

n

=

2 n

X

k,l=1

hε

(k−1)2

−n

, δ

_l2

−n

i

_H

.

(3.47)

Then, underassumption (

H

q

), wehave the following estimates:

|A

n

| ≤ C2

2Hqn+2n

(α

n

)

2q

,

|B

n

| + |C

n

| ≤ C2

2Hqn

(α

n

)

2q−1

γ

n

,

|D

n

| ≤ C2

2Hqn

q

X

r=1

(α

n

)

2q−2r

β

r,n

,

where

C

isa onstantdependingonlyon

q

andthefun tion

f

. Noti ethatthese ondinequality follows from the fa tthat when

(a, b) 6= (q, q)

, or

(a, b) = (q, q)

and

c + d + e + f = 2q

with

d ≥ 1

or

e ≥ 1

,there will be at least a fa tor of theform

hε

(k−1)2

−n

, δ

_l2

−n

i

_H

intheexpression of

B

n

or

C

n

.

Inthe ase

H <

1

2

,wehaveby(2.34 )that

α

n

≤ 2

−2nH

,by(2.36 )that

β

r,n

≤ C2

n−2rHn

, andby(2.35 )that

γ

n

≤ C2

n

. Asa onsequen e,we obtain

|A

n

| ≤ C2

n(−2Hq+2)

,

(3.48)

|B

n

| + |C

n

| ≤ C2

n(−2Hq+2H+1)

,

(3.49)

|D

n

| ≤ C

q

X

r=1

2 n(−2(q−r)H+1)

,

(3.50)

(16)

In the ase

1

2 ≤ H < 1 −

2q

1

, we have by (2.38 ) that

α

n

≤ C2

−n

, by (2.36 ) that

β

r,n

≤ C2

n−2rHn

,andby(2.35 )that

γ

n

≤ C2

n

. Asa onsequen e,we obtain

|A

n

| + |B

n

| + |C

n

| ≤ C2

n(2q(H−1)+2)

,

|D

n

| ≤ C

q

X

r=1

2 n((2q−2r)(H−1)+1)

,

whi h also implies(3.42). Finally, if

H = 1 −

1 2q

, we have by (2.38 ) that

α

n

≤ C2

−n

, by (2.37 ) that

β

r,n

≤

Cn2

2n−2rn

,and by(2.35 )that

γ

n

≤ C2

n

. Asa onsequen e, we obtain

|A

n

| + |B

n

| + |C

n

| ≤ C2

n

,

|D

n

| ≤ C

q

X

r=1

n2

n

q

r

,

whi h implies(3.43 ).

3.1 Proof of Theorem 1 in the ase

0 < H <

1 2q

Inthis subse tion we aregoing to prove the rst point of Theorem 1. The proof will be done in three steps. Set

V

(q)

1,n

(f ) = 2

n(qH−1)

V

(q)

n

(f )

. We rst study the asymptoti behavior of

E

V

_1,n

(q)

(f )

2

,using Proposition8.

Step 1. Thede omposition(3.44 )leads to

E

V

_1,n

(q)

(f )

2 = 2

2n(qH−1)

(A

n

+ B

n

+ C

n

+ D

n

) .

From the estimate (3.49 ) we obtain

2 2n(qH−1)

_(|B

n

| + |C

n

|) ≤ C2

n(2H−1)

,

whi h onverges to

zeroas

n

goesto innitysin e

H <

1 2q

<

1

2

. Ontheother hand(3.50 )yields

2 2n(qH−1)

_|D

n

| ≤ C

q

X

r=1

2 n(2rH−1)

,

(17)

2 4Hqn−2n

q!

2

2 n

X

k,l=1

E

n

f

(q)

(B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

o

hε

(k−1)2

−n

, δ

_k2

−n

i

q

H

hε

(l−1)2

−n

, δ

l2

−n

i

q

H

−

2 −2n−2q

_q!

₂

2 n

X

k,l=1

E

n

f

(q)

(B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

o

≤ C2

2Hn−n

_,

whi h implies, as

n → ∞

:

E V

_1,n

(q)

(f )

2 =

2 −2n−2q

q!

2

2 n

X

k,l=1

E

n

f

(q)

(B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

o

+ o(1).

(3.52)

Step 2: Weneed theasymptoti behaviorof thedoubleprodu t

J

n

:= E

V

1,n

(q)

(f ) × 2

−n

2 n

X

l=1

f

(q)

(B

_(l−1)2

−n

)

!

.

Usingthesame argumentsasinStep1 we obtain

J

n

= 2

Hqn−2n

2 n

X

k,l=1

E

n

f (B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

) H

_q

2 nH

∆B

_k2

−n

o

=

1 q!

2 2Hqn−2n

2 n

X

k,l=1

E

n

f (B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

) I

_q

δ

⊗q

k2

−n

o

=

1 q!

2 2Hqn−2n

2 n

X

k,l=1

E

nD

D

q

f (B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

, δ

_k2

⊗q

−n

E

H

⊗q

o

= 2

2Hqn−2n

2 n

X

k,l=1

q

X

a=0

1 a!(q − a)!

E

n

f

(a)

(B

_(k−1)2

−n

) f

(2q−a)

(B

_(l−1)2

−n

)

o

×hε

(k−1)2

−n

, δ

_k2

−n

i

_H

a

hε

_(l−1)2

−n

, δ

_k2

−n

i

q−a

H

.

It turns out thatonly the term with

a = q

will ontribute to the limit as

n

tends to innity. For thisreason we make thede omposition

(18)

|S

n

| ≤ C2

2Hn−n

,

whi h tends tozero as

n

goesto innity. Moreover, by(3.51 ), we have

2 2Hqn−2n

q!

2 n

X

k,l=1

E

n

f

(q)

(B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

o

hε

(k−1)2

−n

, δ

_k2

−n

i

q

_H

−(−1)

q

2 −2n−q

_q!

2 n

X

k,l=1

E

n

f

(q)

(B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

o

≤ C 2

2Hn−n

_,

whi h also tendsto zeroas

n

goesto innity. Thus,nally, as

n → ∞

:

J

n

= (−1)

q

2 −2n−q

q!

2 n

X

k,l=1

E

n

f

(q)

(B

_(k−1)2

−n

) f

(q)

(B

_(l−1)2

−n

)

o

+ o(1).

(3.53)

Step 3: By ombining (3.52 )and (3.53 ),weobtain that

E

V

(q)

1,n

(f ) −

(−1)

q

2 q

_q!

2 −n

2 n

X

k=1

f

(q)

(B

_(k−1)2

−n

)

2 = o(1),

as

n → ∞

. Thus, the proof of the rst point of Theorem 1 is done using a Riemann sum

argument.

3.2 Proof of Theorem 1 in the ase

H > 1

−

1 2q

: the weighted non- entral limit theorem

Weprove here thatthesequen e

V

3,n

(f )

,given by

V

_3,n

(q)

(f ) = 2

n(1−H)q−n

V

_n

(q)

(f ) = 2

qn−n

1 q!

2 n

X

k=1

f B

_(k−1)2

−n

I

q

δ

_k2

⊗q

−n

,

onverges in

L

2

as

n → ∞

to the pathwiseintegral

R

1

0 f (B

s

)dZ

(q)

s

withrespe tto theHermite pro ess of order

q

introdu ed inDenition 7.

Observerstthat,by onstru tionof

Z

(q)

(pre isely,seethedis ussionbeforeDenition 7 in Se tion 2), the desired result is in order when the fun tion

f

is identi ally one. More pre isely:

Lemma 9 For ea h xed

t ∈ [0, 1]

, the sequen e

2 qn−n 1

q!

P

[2

n

_t]

k=1

I

q

δ

_k2

⊗q

−n

onverges in

L

2

to the Hermite random variable

Z

(19)

Now, onsider the ase of a general fun tion

f

. We x two integers

m ≥ n

, and de omposethe sequen e

V

(q)

3,m

(f )

asfollows:

V

_3,m

(q)

(f ) = A

(m,n)

+ B

(m,n)

,

where

A

(m,n)

=

1 q!

2 m(q−1)

2 n

X

j=1

f B

_(j−1)2

−n

j2

X

m−n

i=(j−1)2

m−n

₊₁

I

q

δ

_i2

⊗q

−m

,

and

B

(m,n)

=

1 q!

2 m(q−1)

2 n

X

j=1

j2

m−n

X

i=(j−1)2

m−n

₊₁

∆

m,n

_i,j

f (B) I

q

δ

_i2

⊗q

−m

,

withthenotation

∆

m,n

i,j

f (B) = f (B

(i−1)2

−m

) − f(B

_(j−1)2

−n

)

. Weshallstudy

A

(m,n)

and

B

(m,n)

separately. Study of

A

(m,n)

. When

n

isxed, Lemma 9 yields thattherandom ve tor





1 q!

2 m(q−1)

j2

m−n

X

i=(j−1)2

m−n

₊₁

I

q

δ

_i2

⊗q

−m

; j = 1, . . . , 2

n





onverges in

L

2

,as

m → ∞

,tothe ve tor

Z

_j2

(q)

−n

− Z

(q)

(j−1)2

−n

; j = 1, . . . , 2

n

.

Then, as

m → ∞

,

A

(m,n) L

_{→ A}

2 (∞,n)

,where

A

(∞,n)

:=

2 n

X

j=1

f (B

_(j−1)2

−n

)

Z

_j2

(q)

−n

− Z

(q)

(j−1)2

−n

.

Finally, we laim that when

n

tends to innity,

A

(∞,n)

onverges in

L

2

to

R

1

0 f (B

s

) dZ

(q)

s

. Indeed, observe that the sto hasti integral

R

1

0 f (B

s

) dZ

(q)

s

is a pathwise Young integral. So, to getthe onvergen e in

L

2

itsu es to showthat the sequen e

A

(∞,n)

isbounded in

L

p

for

some

p ≥ 2

. The integral

R

1

0 f (B

s

) dZ

(q)

s

hasmomentsof allorders, be ause for all

p ≥ 2

(20)

if

γ < q(H − 1) + 1

and

β < H

. Onthe other hand,Young's inequalityimplies

A

(∞,n)

−

Z

1

0 f (B

s

) dZ

s

(q)

≤ c

ρ,ν

Var

ρ

f (B)

Var

ν

Z

(q)

,

where

Var

ρ

denotes the variation of order

ρ

,andwith

ρ, ν > 1

su hthat

1 ρ

+

1 ν

> 1

. Choosing

ρ >

_H

1

and

ν >

1 q(H−1)+1

,theresultfollows.

This proves that, by letting

m

and then

n

go to innity,

A

(m,n)

onverges in

L

2

to

R

1

0 f (B

s

) dZ

(q)

s

.

Study of the term

B

(m,n)

: We provethat

lim

n→∞

sup

m

E

B

(m,n)

2 = 0.

(3.54)

Wehave, usingthe produ tformula(2.32) for multiplesto hasti integrals,

E

_B

(m,n)

2 = 2

2m(q−1)

2 n

X

j=1

j2

m−n

X

i=(j−1)2

m−n

₊₁

2 n

X

j

′

₌₁

j

′

₂

m−n

X

i

′

_=(j

′

₋₁₎₂

m−n

₊₁

q

X

l=0

l!

q!

2 q

l

2 ×b

(m,n)

l

hδ

i2

−m

, δ

_i

′

₂

−m

i

l

_H

,

(3.55) where

b

(m,n)

_l

= E

∆

m,n

_i,j

f (B)∆

m,n

_i

′

_,j

′

f (B)I

_2(q−l)

δ

_i2

⊗(q−l)

−m

⊗δ

e

_i

⊗(q−l)

′

₂

−m

.

(3.56)

By(2.31) and (2.30 ),we obtain that

b

(m,n)

l

isequal to

E

D

2(q−l)

∆

m,n

_i,j

f (B)∆

m,n

_i

′

_,j

′

f (B)

, δ

_i2

⊗(q−l)

−m

⊗δ

e

_i

⊗(q−l)

′

₂

−m

E

H

⊗2(q−l)

=

2q−2l

_X

a=0

2q − 2l

a

D

E

f

(a)

(B

_(i−1)2

−m

)ε

⊗a

(i−1)2

−m

− f

(a)

_(B

(j−1)2

−n

)ε

⊗a

(j−1)2

−n

e

⊗

f

(2q−2l−a)

(B

(i

′

₋₁₎₂

−m

)ε

⊗b

(i

′

₋₁₎₂

−m

− f

(2q−2l−a)

(B

(j

′

₋₁₎₂

−n

)ε

⊗b

(j

′

₋₁₎₂

−m

, δ

_i2

⊗(q−l)

−m

⊗δ

e

_i

⊗(q−l)

′

₂

−m

E

H

⊗2(q−l)

.

Theterm in(3.55 ) orrespondingto

l = q

anbe estimatedby

1 q!

2 2m(q−1)

_sup

|x−y|≤2

−n

E |f(B

x

) − f(B

y

)|

2 _β

q,m

,

where

β

q,m

hasbeenintrodu edin(3.46 ). Soit onvergestozeroas

n

tendstoinnity,uniformly in

m

,be ause, by(2.40 )and usingthat

H > 1 −

(21)

Inorderto handle thetermswith

0 ≤ l ≤ q − 1

,we makethe de omposition

b

(m,n)

l

≤

2q−2l

_X

a=0

2q − 2l

a

_X

4 h=1

B

h

,

(3.57) where

B

1 = E

∆

m,n

i,j

f (B)∆

m,n

i

′

_,j

′

f (B)

D

ε

⊗a

_(i−1)2

−m

⊗ε

e

⊗(2q−2l−a)

_(i

′

₋₁₎₂

−m

, δ

_i2

⊗(q−l)

−m

⊗δ

e

_i

⊗(q−l)

′

₂

−m

E

H

⊗2(q−l)

,

B

2 = E

f

(a)

(B

_(j−1)2

−n

)∆

m,n

_i

′

_,j

′

f (B)

×

D

ε

⊗a

(i−1)2

−m

− ε

⊗a

_(j−1)2

−n

e

⊗ε

⊗(2q−2l−a)

_(i

′

₋₁₎₂

−m

, δ

_i2

⊗(q−l)

−m

⊗δ

e

⊗(q−l)

_i

′

₂

−m

E

H

⊗2(q−l)

,

B

3 = E

∆

m,n

i,j

f (B)f

(2q−2l−a)

(B

(j

′

₋₁₎₂

−n

)

×

D

ε

⊗a

_(i−1)2

−m

⊗

e

ε

⊗(2q−2l−a)

_(i

′

₋₁₎₂

−m

− ε

⊗(2q−2l−a)

_(j

′

₋₁₎₂

−n

, δ

⊗(q−l)

_i2

−m

⊗δ

e

_i

⊗(q−l)

′

₂

−m

E

H

⊗2(q−l)

,

B

4 = E

f

(a)

(B

_(j−1)2

−n

)f

(2q−2l−a)

(B

_(j

′

₋₁₎₂

−n

)

×

D

ε

⊗a

_(i−1)2

−m

− ε

⊗a

_(j−1)2

−n

e

⊗

ε

⊗(2q−2l−a)

_(i

′

₋₁₎₂

−m

− ε

⊗(2q−2l−a)

_(j

′

₋₁₎₂

−n

, δ

_i2

⊗(q−l)

−m

⊗δ

e

_i

⊗(q−l)

′

₂

−m

E

H

⊗2(q−l)

.

(3.58) By using (2.38 ) and the onditions imposed on the fun tion

f

,one an bound the terms

B

1

,

B

2

and

B

3

asfollows:

|B

1 | ≤ c(q, f, H)

sup

|x−y|≤

2n

1 ,0≤a≤2q

E

f

(a)

(B

x

) − f

(a)

(B

y

)

2

2 −2m(q−l)

,

|B

2 | + |B

3 | ≤ c(q, f, H)

sup

|x−y|≤

2n

1 ,0≤a≤2q

E

_f

(2q−2l−a)

(B

x

) − f

(2q−2l−a)

(B

y

)

2 −2m(q−l)

,

and,byusing (2.39 ),weobtain that

(22)

E

_B

(m,n)

2 ≤ R

n

+ c(H, f, q)2

2m(q−1)

sup

|x−y|≤

1 2n

,0≤a≤2q

f

(2q−2l−a)

(B

x

) − f

(2q−2l−a)

(B

y

)

+ (2

−n

)

q−1

q

!

×

2 n

X

j=1

j2

m−n

X

i=(j−1)2

m−n

₊₁

2 n

X

j

′

₌₁

j

′

₂

m−n

X

i

′

_=(j

′

₋₁₎₂

m−n

₊₁

q−1

X

l=0

2 −2m(q−l)

_hδ

_i2

−m

, δ

_i

′

₂

−m

i

l

_H

≤ R

n

+ c(H, f, q)2

2m(q−1)

sup

|x−y|≤

1 2n

,0≤a≤2q

f

(2q−2l−a)

(B

x

) − f

(2q−2l−a)

(B

y

)

+ (2

−n

)

q−1

q

!

×

q−1

X

l=0

2 −2m(q−l)

2 m

X

i,j=0

hδ

i2

−m

, δ

_i

′

₂

−m

i

l

_H

≤ R

n

+ c(H, f, q)

sup

|x−y|≤

2n

1 ,0≤a≤2q

f

(2q−2l−a)

(B

x

) − f

(2q−2l−a)

(B

y

)

+ (2

−n

)

q−1

q

!

andthis onverges to zerodue to the ontinuity of

B

andsin e

q > 1

. 3.3 Proof of Theorem 1 in the ase

1 2q

< H

≤ 1 −

1 2q

: the weighted entral limit theorem

Suppose rst that

1 2q

< H < 1 −

2q

1

. We study the onvergen e in law of the sequen e

V

_2,n

(q)

(f ) = 2

−

n

2 V

_n

(q)

(f )

. We xtwo integers

m ≥ n

,and de omposethis sequen e asfollows:

V

_2,m

(q)

(f ) = A

(m,n)

+ B

(m,n)

,

where

A

(m,n)

= 2

−

m

2

2 n

X

j=1

f B

_(j−1)2

−n

j2

X

m−n

i=(j−1)2

m−n

₊₁

H

q

2 mH

∆B

i2

−m

,

and

B

(m,n)

=

1 q!

2 m(Hq−

1

2 )

2 n

X

j=1

j2

m−n

X

i=(j−1)2

m−n

₊₁

∆

m,n

_i,j

f (B)I

q

δ

⊗q

_i2

−m

,

andwhere asbeforewe makeuseof thenotation

∆

m,n

i,j

f (B) = f (B

(i−1)2

−m

) − f(B

_(j−1)2

−n

)

.

Letus rst onsider the term

A

(m,n)

. From Theorem 1 in Breuer and Major [1℄, and takinginto a ount that

H < 1 −

1 2q

,itfollows thattherandom ve tor