Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H=1/4

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Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H=1/4

Ivan Nourdin, Anthony Réveillac

To cite this version:

Ivan Nourdin, Anthony Réveillac. Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H=1/4. 2008. �hal-00258524�

hal-00258524, version 1 - 22 Feb 2008

Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: the critical case H = 1/4

Ivan Nourdin¹ and Anthony R´eveillac²

Abstract: We derive the asymptotic behavior of weighted quadratic variations of fractional Brow- nian motion B with Hurst index H = 1/4. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C.A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated toB.

Key words: Fractional Brownian motion; quartic process; change of variable formula; weighted quadratic variations; Malliavin calculus; weak convergence.

Present version: February 2008

1 Introduction

Let B^H be a fractional Brownian motion with Hurst index H ∈ (0,1). Since the seminal works by Breuer and Major [1], Dobrushin and Major [4], Giraitis and Surgailis [5] or Taqqu [24], it is well-known that

• ifH ∈(0,³₄) then

√1 n

n−1X

k=0

n^2H(B_(k+1)/n^H −B_k/n^H )²−1 Law

−−−→n→∞

N(0, C_H²); (1.1)

• ifH = ³₄ then

√ 1 n logn

n−1X

k=0

n^3/2(B_(k+1)/n^3/4 −B_k/n^3/4)²−1 Law

−−−→_n→∞ N (0, C²3 4

); (1.2)

• ifH ∈(³₄,1) then n^1−2H

n−1X

k=0

n^2H(B_(k+1)/n^H −B_k/n^H )²−1 Law

−−−→_n→∞ “Rosenblatt r.v.”. (1.3)

1Universit´e Pierre et Marie Curie Paris VI, Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Boˆite courrier 188, 4 place Jussieu, 75252 Paris Cedex 05, France,[email protected]

2Universit´e de La Rochelle Laboratoire Math´ematiques, Image et Applications, Avenue Michel Cr´epeau, 17042 La Rochelle Cedex, France[email protected]

Here,C_H >0 denotes a constant depending only onHand which can be computed explicitly.

Moreover, the term “Rosenblatt r.v.” denotes a random variable whose distribution is the same as that of the Rosenblatt processZ at time one, see (1.9) below.

Now, letf be a real function assumed to be regular enough. Very recently, the asymptotic behavior of

n−1X

k=0

f(B_k/n^H )

n^2H(B_(k+1)/n^H −B_k/n^H )²−1

(1.4) received a lot of attention, see [6, 12, 13, 14, 16] (see also the related works [17, 20, 21, 23]).

The initial motivation of such a study was to derive the exact rates of convergence of some approximation schemes associated to scalar stochastic differential equations driven by B^H, see [6, 12, 13] for precise statements. But it turned out that it was also interesting for itself because it highlighted new phenomena with respect to (1.1)-(1.3). Indeed, in the study of the asymptotic behavior of (1.4), a new critical value (H= ¹₄) has appeared. More precisely:

• ifH < ¹₄ then n^2H−1

n−1X

k=0

f(B_k/n^H )

n^2H(B^H_(k+1)/n−B_k/n^H )²−1 L²

−−−→_n→∞ 1 4

Z 1

0

f^′′(B_s^H)ds; (1.5)

• if ¹₄ < H < ³₄ then

√1 n

n−1X

k=0

f(B_k/n^H )

n^2H(B_(k+1)/n^H −B_k/n^H )²−1 Law

−−−→_n→∞ C_H Z ₁

0

f(B_s^H)dW_s (1.6) forW a standard Brownian motion independent of B^H;

• ifH = ³₄ then

√ 1 n logn

n−1X

k=0

f(B_k/n^3/4)

n^3/2(B_(k+1)/n^3/4 −B_k/n^3/4)²−1 Law

−−−→_n→∞ C_3/4 Z ₁

0

f(B_s^3/4)dW_s (1.7) forW a standard Brownian motion independent of B^3/4;

• ifH > ³₄ then n^1−2H

n−1X

k=0

f(B^H_k/n)

n^2H(B_(k+1)/n^H −B_k/n^H )²−1 L²

−−−→_n→∞

Z ₁

0

f(B_s^H)dZ_s (1.8) forZ the Rosenblatt process defined by

Z_s =I₂^X(L_s), (1.9)

whereI₂^X denotes the double stochastic integral with respect to the Wiener processX given by the transfer equation (2.3) and where, for everys∈[0,1],Lsis the symmetric square integrable kernel given by

L_s(y₁, y₂) = 1

21_[0,s]²(y₁, y₂) Z s

y1∨y2

∂K_H

∂u (u, y₁)∂K_H

∂u (u, y₂)du.

Even if it is not completely obvious at first glance, convergences (1.1) and (1.5) well agree. Indeed, since 2H−1<−¹₂ if and only ifH < ¹₄, (1.5) is actually a particular case of (1.1) when f ≡1. The convergence (1.5) is proved in [14] while the other cases (1.6)-(1.8) are proved in [16]. On the other hand, notice that the relations (1.5)-(1.8) do not cover the critical case H = ¹₄. Our first main result completes this important (see why just below) missing case:

Theorem 1.1. If H = ¹₄ then

√1 n

n−1X

k=0

f(B_k/n^1/4)√

n(B_(k+1)/n^1/4 −B_k/n^1/4)²−1 Law

−−−→_n→∞ C_1/4 Z 1

0

f(B_s^1/4)dWs+1 4

Z 1

0

f^′′(B^1/4_s )ds (1.10) for W a standard Brownian motion independent of B^1/4 and where

C_1/4² = 1 2

X∞

p=−∞

p|p+ 1|+p

|p−1| −2p

|p|2

<∞.

Here, it is interesting to compare the obtained limit in (1.10) with those obtained in the recent work [17]. In [17], the authors also studied the asymptotic behavior of (1.4) but when the fractional Brownian motion B^H is replaced by an iterated Brownian motion Z, that is the process defined byZ_t=X(Y_t),t∈[0,1], withXandY two independent standard Brownian motions. Iterated Brownian motionZ is self-similar of index ¹₄ and has stationary increments. Thus, although if it is not Gaussian, Z is “close” to the fractional Brownian motion B^1/4. For Z instead of B^1/4, it is proved in [17] that the correctly renormalized weighted quadratic variation (which is note exactly defined as in (1.4), but rather by means of a random partition composed of Brownian hitting times) converges in law towards the so-called weighted Brownian motion in random sceneryat time one, defined as

√2 Z +∞

−∞

f(Xx)L^x₁(Y)dWx,

compare with the right-hand side of (1.10). Here, {L^x_t(Y)}x∈R,t∈[0,1] stands for the jointly continuous version of the local time process of Y, while W denotes a two-sided standard Brownian motion independent of X and Y.

For now, we takeB^H =B^1/4to be a fractional Brownian motion with Hurst indexH= ¹₄. This particular value ofH is important because the fractional Brownian motion with Hurst indexH = ¹₄ has a remarkable physical interpretation in terms of particle systems. Indeed, if one consider an infinite number of particles, initially placed on the real line according to a Poisson distribution, performing independent Brownian motions and undergoing “elastic”

collisions, then the trajectory of a fixed particle (after rescaling) converges to a fractional Brownian motion with Hurst index H= ¹₄. This striking fact has been first pointed out by Harris in [10], and then rigorously proven in [3] (see also references therein).

Now, let us explain an interesting consequence of a slight modification of Theorem 1.1 towards a first step in the construction of a stochastic calculus with respect to B^1/4. As it is nicely explained by Swanson in [23], there are at least two kinds of Stratonovitch-type

Riemann sums that one can consider in order to defineR₁

0 f(Bs^1/4)◦dB^1/4s whenf is a real smooth function. The first corresponds to the so-called “trapezoid rule” and is given by

S_n(f) =

n−1X

k=0

f(B_k/n^1/4) +f(B_(k+1)/n^1/4 )

2 B_(k+1)/n^1/4 −B_k/n^1/4 .

The second corresponds to the so-called “midpoint rule” and is given by Tn(f) =

⌊n/2⌋

X

k=1

f(B_(2k−1)/n^1/4 ) B^1/4_(2k)/n−B_(2k−2)/n^1/4 .

By Theorem 3 in [16] (see also [2, 7, 8]), we have that Z ₁

0

f^′(B_s^1/4)d^◦B_s^1/4:= lim

n→∞S_n(f^′) exists in probability and verifies the following classical change of variable formula:

Z 1

0

f^′(B_s^1/4)d^◦B_s^1/4 =f(B₁^1/4)−f(0). (1.11) On the other hand, it is quoted in [23] that Burdzy and Swanson conjectured³ that

Z 1

0

f^′(B_s^1/4)d^⋆B^1/4_s := lim

n→∞T_n(f^′) exists in law

and verifies, this time, the following non classical change of variable formula:

Z ₁

0

f^′(B_s^1/4)d^⋆B_s^{1/4 Law}= f(B₁^1/4)−f(0)−κ 2

Z ₁

0

f^′′(B_s^1/4)dW_s, (1.13) whereκ is an explicit universal constant andW denotes a standard Brownian motion inde- pendent of B^1/4. Our second main result is the following:

3In reality, Burdzy and Swanson conjectured (1.13) not for the fractional Brownian motionB^1/4 but for processF defined by

Ft=u(t,0), t∈[0,1], (1.12)

where

ut= 1

2uxx+ ˙W(t, x), t∈[0,1], x∈R, with initial conditionu(0, x) = 0.

(Here, as usual, ˙W denotes the space-time white noise on [0,1]×R). It is immediately checked thatF is a centered Gaussian process with covariance function

E(FsFt) = 1

√2π

√t+s−p

|t−s| .

so thatF is actually abifractional Brownian motionof indices ¹₂ and ¹₂ in the sense of Houdr´e and Villa [9].

Using the main result of [11], we have that B^1/4 and F actually differ only from a process with absolutely continuous trajectories. As a direct consequence, using a Girsanov type transformation, we immediately see that it is equivalent to prove (1.13) either forB^1/4 or forF.

As quoted in [22, Remark 12], notice finally that the change of variable formula (1.11) also holds forF.

Theorem 1.2. The conjecture of Burdzy and Swanson is true. More precisely, (1.13) holds for any real function f :R→R verifying(H₉) (see Section 3 below).

Finally, we would mention that the strategy we used in this paper can also be derived in order to obtain the following analogue of Theorem 1.2, that we propose (in order to keep the length of the present paper within limit) to prove in a forthcoming paper:

Theorem 1.3. Let f :R→R be smooth enough. ThenR₁

0 f^′(Bs^1/6)d^◦Bs^1/6 exists in law and verifies

Z ₁

0

f^′(B_s^1/6)d^◦B_s^1/6Law= f(B₁^1/6)−f(0)−eκ 6

Z ₁

0

f^′′′(B_s^1/6)dW_s,

where eκ is an explicit universal constant and W denotes a standard Brownian motion inde- pendent of B^1/6.

The rest of the paper is organized as follows. In Section 2, we recall some notion con- cerning fractional Brownian motion. In Section 3, we prove Theorem 1.1. In Section 4, we prove Theorem 1.2.

2 Preliminaries and notations

We begin by briefly recalling some basic facts about stochastic calculus with respect to a fractional Brownian motion. We refer to [18] for further details. Let B^H = (B_t^H)_t∈[0,1] be a fractional Brownian motion with Hurst parameter H∈(0,¹₂) defined on a probability space (Ω,A, P). We mean thatB^H is a centered Gaussian process with the covariance function

R_H(s, t) = 1

2 t^2H+s^2H− |t−s|^2H

. (2.1)

We denote by E the set of step R−valued functions on [0,1]. Let H be the Hilbert space defined as the closure of E with respect to the scalar product

1_[0,t],1_[0,s]

H=R_H(t, s).

The covariance kernelR_H(t, s) introduced in (2.1) can be written as R_H(t, s) =

Z s∧t

0

K_H(s, u)K_H(t, u)du, where K_H(t, s) is the square integrable kernel defined by

KH(t, s) =cH

"

t s

H−¹₂

(t−s)^H−12 −(H− 1 2)s^12−H

Z t

s

u^H−32(u−s)^H−12du

#

, 0< s < t, (2.2) where c_H² = 2H(1−2H)⁻¹β(1−2H, H+ 1/2)⁻¹ and β denotes the Beta function. By convention, we setK_H(t, s) = 0 if s≥t.

LetK^∗H :E →L²([0,1]) be the linear operator defined by:

K^∗H 1_[0,t]

=KH(t,·).

The following equality holds for any s, t∈[0,1]:

h1_[0,t],1_[0,s]i^H=hK^∗H1_[0,t],K^∗H1_[0,s]iL²([0,1])= E B_t^HB_s^H

and then K^∗_H provides an isometry between the Hilbert spaces H and a closed subspace of L²([0,1]). The processX = (X_t)_t∈[0,1] defined by

X_t=B^H (K^∗H)⁻¹(1_[0,t])

(2.3) is a Wiener process, and the process B^H has an integral representation of the form

B_t^H = Z _t

0

K_H(t, s)dX_s.

LetS be the set of all smooth cylindrical random variables,i.e. of the form

F =ψ(B_t^H₁, . . . , B^H_tm) (2.4) where m≥1,ψ:R^m→R∈C^∞

b and 0≤t₁< . . . < t_m≤1. The Malliavin derivative of F with respect to B^H is the element of L²(Ω,H) defined by

DsF = Xm

i=1

∂ψ

∂x_i(B_t^H₁, . . . , B_t^H_m)1_[0,ti](s), s∈[0,1].

In particularD_sB_t^H =1_[0,t](s). For any integer k≥1, we denote byD^k,2 the closure of the set of smooth random variables with respect to the norm

kFk²k,2 = E F²

+ Xk

j=1

E

|D^jF|²H^⊗j

.

The Malliavin derivative Dverifies the chain rule: if ϕ:Rⁿ→R isC¹

b and if (F_i)_i=1,...,n is a sequence of elements of D^1,2 thenϕ(F₁, . . . , F_n)∈D^1,2 and we have, for anys∈[0,1]:

Dsϕ(F1, . . . , Fn) = Xn

i=1

∂ϕ

∂x_i(F1, . . . , Fn)DsFi.

The divergence operatorI is the adjoint of the derivative operator D. If a random variable u∈L²(Ω,H) belongs to the domain of the divergence operator, that is if it verifies

|EhDF, ui^H| ≤c_ukFkL² for any F ∈S, thenI(u) is defined by the duality relationship

E F I(u)

= E hDF, ui^H , for everyF ∈D^1,2.

For every n ≥ 1, let Hn be the nth Wiener chaos of B^H, that is, the closed linear sub- space of L²(Ω,A, P) generated by the random variables {H_n B^H(h)

, h∈ H,|h|H = 1}, whereH_n is thenth Hermite polynomial. The mappingI_n(h^⊗n) =n!H_n B^H(h)

provides a linear isometry between the symmetric tensor product H^⊙n and Hn. For H = ¹₂, I_n coincides with the multiple stochastic integral. The following duality formula holds

E(F I_n(h)) =E hDⁿF, hiH^⊗n

, (2.5)

for any elementh∈H^⊙nand any random variableF ∈D^n,2. Let{e_k, k≥1}be a complete orthonormal system in H. Given f ∈H^⊙p and g∈ H^⊙q, for every r = 0, . . . , p∧q, the rth contraction of f and g is the element ofH^{⊗(p+q−2r)} defined as

f ⊗rg= X∞

i1,...,ir=1

hf, e_i1 ⊗. . .⊗e_iriH^⊗r ⊗ hg, e_i1 ⊗. . .⊗e_iriH^⊗r.

Note that f ⊗0g =f ⊗g equals the tensor product off and g while, for p =q, f⊗pg = hf, giH^⊗p. Finally, we mention the useful following multiplication formula: if f ∈ H^⊙p and g∈H^⊙q, then

I_p(f)I_q(g) = Xp∧q

r=0

r!

p r

q r

I_p+q−2r(f⊗rg). (2.6)

3 Proof of Theorem 1.1

In all this section, B =B^1/4 denotes a fractional Brownian motion with Hurst index H = 1/4.

Let

G_n := 1

√n

n−1X

k=0

f(B_k/n)[√

n(B_(k+1)/n−B_k/n)²−1], n≥1.

For k= 0, . . . , n−1 andt∈[0,1], we set

δ_k/n :=1[k/n,(k+1)/n] and ε_t:=1_[0,t].

The relations between Hermite polynomials and multiple stochastic integrals (see Section 2) allow to write

√n(B_(k+1)/n−B_k/n)²−1 =√

n I₂(δ_k/n^⊗2 ).

As a consequence:

G_n=

n−1X

k=0

f(B_k/n)I₂(δ_k/n^⊗2).

In the sequel, for f :R→R, we will need assumption of the type:

Hypothesis (H_q):

The function f :R→Rbelongs to C^q and is such that sup

t∈[0,1]

E

|f⁽ⁱ⁾(B_t)|^p

<∞

for any p≥1 and i∈ {0, . . . , q}.

We begin by the following technical lemma:

Lemma 3.1. Let n≥1 andk= 0, . . . , n−1. We have (i) |E(B_r(B_t−B_s))| ≤√

t−sfor any r∈[0,1] and 0≤s < t≤1, (ii) sup

t∈[0,1]

n−1X

k=0

ε_t, δ_k/n

H

=

n→∞O(1),

(iii)

n−1X

k,j=0

ε_j/n, δ_k/n

H

=

n→∞O(n),

(iv)

ε_k/n, δ_k/n2 H− 1

4n ≤

√k+ 1−√ k

4n ; consequently

n−1X

k=0

ε_k/n, δ_k/n2 H− 1

4n

_n→∞−→ 0.

Proof of Lemma 3.1.

(i) We have

E(B_r(B_t−B_s)) = 1 2(√

t−√ s) +1

2

p|s−r| −p

|t−r| . Using the classical inequality

p

|b| −p

|a| ≤p

|b−a|, the desired result follows.

(ii) Observe that ε_t, δ_k/n

H= 1 2√n

√

k+ 1−√ k−p

|k+ 1−nt|+p

|k−nt| .

Consequently, we have

n−1X

k=0

ε_t, δ_k/n

H

≤ 1 2+ 1

2√n





⌊nt⌋−1X

k=0

√nt−k−√

nt−k−1

+p

⌊nt⌋+ 1−nt−p

nt− ⌊nt⌋+

n−1X

k=⌊nt⌋+1

√nt−k−√

nt−k−1



.

The desired conclusion follows easily.

(iii) It is a direct consequence of(ii):

n−1X

k,j=0

ε_j/n, δ_k/n

H

≤ n sup

j=0,...,n−1 n−1X

k=0

ε_j/n, δ_k/n

H

n→∞= O(n).

(iv) We have

ε_k/n, δ_k/n2 H− 1

4n = 1

4n √

k+ 1−√ k √

k+ 1−√ k−2

.

Thus, the desired bound is immediately checked by using 0 ≤ √

x+ 1−√ x ≤ 1 available forx≥0.

The main result of the current section is the following:

Theorem 3.2. Under Hypothesis (H₄), we have G_nn→∞−→^Law C_1/4

Z ₁

0

f(B_s)dW_s+1 4

Z ₁

0

f^′′(B_s)ds,

where W = (W_t)_t∈[0,1] is a standard Brownian motion independent of B and

C_1/4:=

vu ut1

2 X∞

p=−∞

p|p+ 1|+p

|p−1| −2p

|p|2

<∞.

Proof. This proof is mainly inspired by the first draft of [15]. During all the proof,C will denote a constant depending only onkf^(a)k∞,a= 0,1,2,3,4, which can differ from one line to another.

Step 1.- We begin the proof by showing the following limits:

n→∞lim E(G_n) = 1 4

Z 1

0

E f^′′(B_s)

ds, (3.1)

and

n→∞lim E G²_n

=C_1/4² Z ₁

0

E f²(B_s)

ds+ 1 16E

Z 1

0

f^′′(B_s)ds 2

. (3.2)

Proof of (3.1): we can write E(G_n) =

n−1X

k=0

E

f(B_k/n)I₂(δ^⊗2_k/n)

=

n−1X

k=0

ED

D²(f(B_k/n)), δ_k/n^⊗2 E

H

=

n−1X

k=0

E f^′′(B_k/n) ε_k/n, δ_k/n2 H

= 1

4n

n−1X

k=0

E f^′′(B_k/n) +

n−1X

k=0

E f^′′(B_k/n)

ε_k/n, δ_k/n2 H− 1

4n

n→∞−→

1 4

Z 1

0

E f^′′(Bs)

ds, by Lemma 3.1(iv) and under (H4).

Proof of (3.2):

By the multiplication formula (2.6), we have

I2(δ^⊗2_j/n)I2(δ^⊗2_k/n) =I4(δ^⊗2_j/n⊗δ_k/n^⊗2) + 4I2(δ_j/n⊗δ_k/n)

δ_j/n, δ_k/n

H+ 2

δ_j/n, δ_k/n2

H. (3.3) Thus

E G²_n

=

n−1X

j,k=0

E

f(B_j/n)f(B_k/n)I₂(δ^⊗2_j/n)I₂(δ_k/n^⊗2 )

=

n−1X

j,k=0

E

f(B_j/n)f(B_k/n)I₄(δ^⊗2_j/n⊗δ^⊗2_k/n)

+ 4

n−1X

j,k=0

E f(B_j/n)f(B_k/n)I₂(δ_j/n⊗δ_k/n) δ_j/n, δ_k/n

H

+ 2

n−1X

j,k=0

E f(B_j/n)f(B_k/n) δ_j/n, δ_k/n2 H

= A_n+B_n+C_n.

Using Malliavin integration by parts formula (2.5), A_n can be expressed as follows:

A_n =

n−1X

j,k=0

ED

D⁴(f(B_j/n)f(B_k/n)), δ^⊗2_j/n⊗δ^⊗2_k/nE

H^⊗4

= 24

n−1X

j,k=0

X

a+b=4

E

f^(a)(B_j/n)f^(b)(B_k/n) D

ε^⊗a_j/n⊗eε^⊗b_k/n, δ^⊗2_j/n⊗δ_k/n^⊗2E

H^⊗4. In fact, in the previous sum, each term is negligible except

n−1X

j,k=0

E f^′′(B_j/n)f^′′(B_k/n) ε_j/n, δ_j/n2 H

ε_k/n, δ_k/n2 H

= E





"_n−1 X

k=0

f^′′(B_k/n)

ε_k/n, δ_k/n2 H

#2



= E





"

1 4n

n−1X

k=0

f^′′(B_k/n) +

n−1X

k=0

f^′′(B_k/n)

ε_k/n, δ_k/n2 H− 1

4n #2



n→∞−→ E 1 4

Z ₁

0

f^′′(B_s)ds 2!

,by Lemma 3.1(iv) and under (H₄).

The other terms appearing inA_nmake no contribution to the limit. Indeed, they have the

form n−1X

j,k=0

E

f^(a)(B_j/n)f^(b)(B_k/n) ε_j/n, δ_k/n

H

Y3

i=1

ε_xi/n, δ_yi/n

H

(where x_i and y_i are for j ork) and, from Lemma 3.1(i) (iii), we have that





supj,k=0,...,n−1

Q₃

i=1

ε_xi/n, δ_yi/n

H

=

n→∞O(n^−3/2), P_n−1

j,k=0

ε_j/n, δ_k/n

H

=

n→∞O(n).

Still using Malliavin integration by parts formula (2.5), we can boundB_n as follows:

|B_n| ≤ 8

n−1X

j,k=0

X

a+b=2

E

f^(a)(B_j/n)f^(b)(B_k/n) D

ε^⊗a_j/n⊗eε^⊗b_k/n, δ_j/n⊗δ_k/nE

H^⊗2

δ_j/n, δ_k/n

H

≤ Cn⁻¹

n−1X

j,k=0

δ_j/n, δ_k/n

H

, by Lemma 3.1(i) and under (H₄)

= Cn^−3/2

n−1X

j,k=0

|ρ(j−k)| ≤Cn^−1/2 X∞

r=−∞

|ρ(r)| =

n→∞O(n^−1/2), where

ρ(r) :=p

|r+ 1|+p

|r−1| −2p

|r|, r∈Z. (3.4)

Observe that the serie P_∞

r=−∞|ρ(r)|is convergent since|ρ(r)| ∼

|r|→∞

1 2|r|⁻³². Finally, we consider the termC_n:

C_n = 1

2n

n−1X

j,k=0

E f(B_j/n)f(B_k/n)

ρ²(j−k)

= 1

2n X∞

r=−∞

(n−1)∧(n−1−r)

X

j=0∨−r

E f(B_j/n)f(B_(j+r)/n) ρ²(r)

n→∞−→

1 2

Z 1

0

E f²(Bs) ds

X∞

r=−∞

ρ²(r) =C_1/4² Z 1

0

E f²(Bs) ds.

The desired convergence (3.2) follows.

Step 2.- Since the sequence (G_n) is bounded inL¹, the sequence G_n,(B_t)_t∈[0,1]

is tight in R×C([0,1]). Assume that G∞,(Bt)_t∈[0,1]

denotes the limit in law of a certain subsequence of G_n,(B_t)_t∈[0,1]

, denoted again by G_n,(B_t)_t∈[0,1]

. We have to prove that

G∞Law

= C_1/4 Z 1

0

f(Bs)dWs+1 4

Z 1

0

f^′′(Bs)ds,

where W denotes a standard Brownian motion independent of B, or equivalently that E

e^iλG∞|(B_t)_t∈[0,1]

= exp

iλ 4

Z ₁

0

f^′′(B_s)ds−λ² 2 C_1/4²

Z ₁

0

f²(B_s)ds

. (3.5) This will be done by showing that for every random variable ξ of the form (2.4) and every real number λ, we have

n→∞lim φ^′_n(λ) =E

e^iλG∞ξ i

4 Z 1

0

f^′′(Bs)ds−λC_1/4² Z 1

0

f²(Bs)ds

(3.6) where

φ^′_n(λ) := d dλE

e^iλGnξ

=iE

G_ne^iλGnξ

, n≥1.

Let us make precise this argument. Because G_∞,(B_t)_t∈[0,1]

is the limit in law of G_n,(B_t)_t∈[0,1]

and (G_n) is bounded inL¹, we have that E(G_∞ξ e^iλG∞) = lim

n→∞E

G_nξ e^iλGn

, ∀λ∈R,

for every ξ of the form (2.4). Furthermore, because convergence (3.6) holds for every ξ of the form (2.4), the conditional characteristic functionλ7→E e^iλG∞|(B_t)_t∈[0,1]

satisfies the following linear ordinary differential equation:

d dλE

e^iλG∞|(B_t)_t∈[0,1]

=E

e^iλG∞|(B_t)_t∈[0,1] i 4

Z ₁

0

f^′′(B_s)ds−λC_1/4² Z ₁

0

f²(B_s)ds

. By solving it, we obtain (3.5), which yields the desired conclusion.

Thus, it remains to show (3.6). By the duality between the derivative and divergence operators, we have

E

f(B_k/n)I₂(δ^⊗2_k/n)e^iλGnξ

=ED D²

f(B_k/n)e^iλGnξ , δ^⊗2_k/nE

H^⊗2

. (3.7)

The first and second derivatives off(B_k/n)e^iλGnξ are given by D

f(B_k/n)e^iλGnξ

=f^′(B_k/n)e^iλGnξ ε_k/n+iλf(B_k/n)e^iλGnξ DGn+f(B_k/n)e^iλGnDξ and

D²

f(B_k/n)e^iλGnξ

= f^′′(B_k/n)e^iλGnξ ε^⊗2_k/n+ 2iλf^′(B_k/n)e^iλGnξ ε_k/n⊗eDG_n +2f^′(B_k/n)e^iλGn ε_k/n⊗eDξ

−λ²f(B_k/n)e^iλGnξ DG^⊗2_n +2iλf(B_k/n)e^iλGn DGn⊗eDξ

+iλf(B_k/n)e^iλGnξD²G_n+f(B_k/n)e^iλGnD²ξ.

Hence, taking expectation and multiplying by δ_k/n^⊗2 yields ED

D²

f(B_k/n)e^iλGnξ , δ_k/n^⊗2E

H^⊗2

= E

f^′′(B_k/n)e^iλGnξ ε_k/n, δ_k/n2

H+ 2iλE

f^′(B_k/n)e^iλGnξ

DG_n, δ_k/n

H ε_k/n, δ_k/n

H

+2E

f^′(B_k/n)e^iλGn

Dξ, δ_k/n

H ε_k/n, δ_k/n

H−λ²E

f(B_k/n)e^iλGnξ

DG_n, δ_k/n2 H

+2iλE

f(B_k/n)e^iλGn

Dξ, δ_k/n

H

DG_n, δ_k/n

H

+iλE

f(B_k/n)e^iλGnξD

D²G_n, δ_k/n^⊗2E

H^⊗2

+E

f(B_k/n)e^iλGnD

D²ξ, δ_k/n^⊗2E

H^⊗2

. (3.8)

We also need explicit expressions for

DG_n, δ_k/n

Hand forD

D²G_n, δ^⊗2_k/nE

H^⊗2. Differentiating G_n we obtain

DGn=

n−1X

l=0

h

f^′(B_l/n)I2(δ_l/n^⊗2)ε_l/n+ 2f(B_l/n)∆B_l/nδ_l/ni

(3.9) and, as a consequence,

DGn, δ_k/n

H =

n−1X

l=0

f^′(B_l/n)I2(δ_l/n^⊗2)

ε_l/n, δ_k/n

H+ 2

n−1X

l=0

f(B_l/n)∆B_l/n

δ_l/n, δ_k/n

H. (3.10) Also

D²G_n =

n−1X

l=0

h

f^′′(B_l/n)I₂(δ_l/n^⊗2)ε^⊗2_l/n+ 4f^′(B_l/n) ∆B_l/n ε_l/n⊗eδ_l/n

+ 2f(B_l/n)δ^⊗2_l/ni ,

and, as a consequence, D

D²G_n, δ^⊗2_k/nE

H^⊗2 =

n−1X

l=0

h

f^′′(B_l/n)I₂(δ_l/n^⊗2)

ε_l/n, δ_k/n2 H

+4f^′(B_l/n) ∆B_l/n

ε_l/n, δ_k/n

H

δ_l/n, δ_k/n

H+ 2f(B_l/n)

δ_l/n, δ_k/n2 H

i . (3.11) Substituting (3.11) into (3.8) yields the following decomposition forφ^′_n(λ) =i E(G_ne^iλGnξ):

φ^′_n(λ) = −2λ

n−1X

k,l=0

E

f(B_k/n)f(B_l/n)e^iλGnξ δ_l/n, δ_k/n2 H

+ i

n−1X

k=0

E

f^′′(B_k/n)e^iλGnξ ε_k/n, δ_k/n2 H+i

n−1X

k=0

r_k,n (3.12) where r_k,n is given by

r_k,n = 2iλE

f^′(B_k/n)e^iλGnξ

DG_n, δ_k/n

H ε_k/n, δ_k/n

H

+2E

f^′(B_k/n)e^iλGn

Dξ, δ_k/n

H ε_k/n, δ_k/n

H−λ²E

f(B_k/n)e^iλGnξ

DG_n, δ_k/n2 H

+2iλE

f(B_k/n)e^iλGn

Dξ, δ_k/n

H

DG_n, δ_k/n

H

+iλ

n−1X

l=0

E

f(B_k/n)e^iλGnξ f^′′(B_l/n)I₂(δ_l/n^⊗2) ε_l/n, δ_k/n2 H

+4iλ

n−1X

l=0

E

f(B_k/n)e^iλGnξf^′(B_l/n) ∆B_l/n ε_l/n, δ_k/n

H

δ_l/n, δ_k/n

H

+E

f(B_k/n)e^iλGnD

D²ξ, δ^⊗2_k/nE

H^⊗2

= X7

j=1

R^(j)_k,n. (3.13)

Remark that the first sum in the right hand side of (3.12) is very similar to C_n presented in Step 1. In fact, similar computations give

n→∞lim −2λ

n−1X

k,l=0

IE[f(B_k/n)f(B_l/n)e^iλGnξ]

δ_l/n, δ_k/n2 H

=−C_1/4² λ Z ₁

0

E

f²(B_s)e^iλG∞ξ

ds. (3.14) Furthermore, the second term of (3.12) is very similar toE(G_n). In fact, using the arguments presented in Step 1, we obtain here that

n→∞lim i

n−1X

k=0

E

f^′′(B_k/n)e^iλGnξ ε_k/n, δ_k/n2 H= i

4 Z ₁

0

E

f^′′(B_s)e^iλG∞ξ

ds. (3.15) Consequently, (3.6) will be shown as soon as we will prove that lim_n→∞P_n−1

k=0r_k,n= 0.This will be done in several steps.

Step 3.- In this step, we state and prove some estimates which will be crucial in the rest of the proof. First, we will show that

E

f^′(B_k/n)f^′(B_l/n)e^iλGnξ I₂(δ_l/n^⊗2)≤ C

n for any 0≤k, l≤n−1. (3.16) Then we will prove that

E

f(B_k/n)f^′(B_j/n)f^′(B_l/n)e^iλGnξ I₄(δ_j/n^⊗2⊗δ_l/n^⊗2) ≤ C

n² for any 0≤k, j, l≤n−1.

(3.17) Proof of (3.16):

Letζ_ξ,k,ndenotes any random variable of the formf^(a)(B_k/n)f^(b)(B_l/n)e^iλGnξ withaand b two positive integers less or equal to four. From the Malliavin integration by parts formula (2.5) we have

E

f^′(B_k/n)f^′(B_l/n)e^iλGnξ I2(δ_l/n^⊗2)

=ED D²

f^′(B_k/n)f^′(B_l/n)e^iλGnξ , δ^⊗2_l/nE

H^⊗2

.

When computing the RHS, three types of terms appear. First, we have some terms of the

form: 









E(ζ_ξ,k,n)

ε_k/n, δ_l/n2 H, or E

ζ_ξ,k,n

Dξ, δ_l/n

H ε_k/n, δ_l/n

H, or E

ζ_ξ,k,nD

D²ξ, δ_l/n^⊗2E

H^⊗2

,

(3.18)

where Dξ and D²ξ are given by:

( Dξ =P_m

i=1 ∂ψ

∂xi(B_t1, . . . , B_tm)ε_ti, D²ξ =P_m

i,j=1 ∂²ψ

∂xj∂xi(B_t1, . . . , B_tm)ε_tj ⊗ε_ti.

From Lemma 3.1 (i) and under (H4), we have that each of the three terms in (3.18) is less or equal toCn⁻¹. The second type of terms we have to deal with is



 E

ζ_ξ,k,n

DG_n, δ_l/n

H ε_k/n, δ_l/n

H, or E

ζ_ξ,k,n

DG_n, δ_l/n

H

Dξ, δ_l/n

H

. (3.19)

By Cauchy-Schwarz inequality, under (H₄) and by using (4.20) in [15], that is

E

DG_n, δ_l/n2 H

≤Cn⁻¹,

we have that both expressions in (3.19) are also less or equal toCn⁻¹. The last type of terms which has to be taken into account is the term

−λ²E

f^′(B_k/n)f^′(B_l/n)e^iλGnξD

D²G_n, δ_k/n^⊗2E

H^⊗2

.

Again, by using Cauchy-Schwarz inequality and the estimate ED

D²Gn, δ^⊗2_k/nE2 H^⊗2

≤Cn⁻²

(which can be obtained by mimicing the proof of (4.20) in [15]), we can conclude that

−λ²E

f^′(B_k/n)f^′(B_l/n)e^iλGnξD

D²G_n, δ_k/n^⊗2E

H^⊗2

≤ C n. As a consequence (3.16) is shown.

Proof of (3.17):

By the Malliavin integration by parts formula (2.5), we have E

ζ_ξ,k,nf^′(B_j/n)f^′(B_l/n)I₄(δ^⊗2_j/n⊗δ^⊗2_l/n)

=ED

D⁴ ζ_ξ,k,nf^′(B_j/n)f^′(B_l/n)

, δ_j/n^⊗2⊗δ^⊗2_l/nE

H^⊗4

.

When computing the RHS, we have to deal with the same type of terms as in the proof of (3.16) plus two additional types of terms containing

ED

D³G_n, δ^⊗2_j/n⊗δ_l/nE2 H^⊗3

and ED

D⁴G_n, δ_j/n^⊗2⊗δ_l/n^⊗2E2 H^⊗4

.