Optimal series representations of continuous Gaussian random ﬁelds

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Antoine AYACHE

Universit´e Lille 1 - Laboratoire Paul Painlev´e

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Organization of the talk

1 Introduction

2 H¨older regularity and rate convergence ofl-numbers Gaussian processes and operators

Proof of the main result of the section

3 The Meyer, Sellan and Taqqu wavelet representations of fBm Representation of Brownian motion in the Faber-Schauder system Wavelet representation of fBm without a scaling function

Wavelet representation with a well-localized scaling function Optimality of both representations

4 Classical series representations of RLp

Some generalities on Riemann-Liouville process Optimality of classical series representations

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Definition 1.1

A random variable X defined on a probability space(Ω,A,P)and with values in a separable Banach space(E,BE), is said to beGaussian centered, when hX,x^∗iis a real-valued centered Gaussian random variablefor any x^∗∈E^∗, the topological dual of E .

Remark 1.1

Let K be a compact metric space and letC(K)be the Banach space of the continuous real-valued functions defined on K . Assume that{X(t)}_t∈K is a real-valued centered Gaussian process, defined on(Ω,A,P)and with continuous paths (for all fixedω∈Ω, the function X(·, ω) :K →R, t7→X(t, ω)is

continuous). Then, the map(Ω,A,P)→(C(K),B_C(K)),ω7→X(·, ω)is a centered Gaussian random variable with values in(C(K),B_C(K)).

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Theorem 1.1

X can be represented as an almost surely convergent random series of the form:

X =

+∞

X

k=1

kαk, (1.1)

where thek’s areindependentN(0,1) real-valued Gaussian random variablesand theαk’s are some deterministic elements ofE .

→See the book of Lifshits (1995) or that of Ledoux and Talagrand (1991) for a proof Theorem 1.1.

→In the case whereX is a Brownian motionB ={B(t)}_t∈[0,1], two classical examples of such series representations are:

the representation in the trigonometric system, B(t) =₀t+√

2

+∞

X

k=1

_ksin(πkt)

πk ;

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tail of the series:

+∞

X

k=n

_kα_k,

tends to zero as fast as possible. This leads to the study of the quantity

l_n(X) = inf





 E

+∞

X

k=n

_kα_k

2!^1/2

; X =

+∞

X

k=1

_kα_k







. (1.2)

l_n(X) is called thenthl-number of X. Remark 1.2

The value of ln(X)remains the same even if the random variablesk are allowed to be dependent.

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Theorem 1.2 (Li and Linde 1999) Let a>0 and b∈Rbe fixed.

1 If for some constant c1>0 and every integer n≥1,

ln(X)≤c1n^−a(1 + logn)^b. (1.3) Then, there is a constant c₂>0 such that for any >0small enough,

log(P(kXk ≤)≥ −c2^−1/a(log 1/)^b/a. (1.4)

2 Conversely, if (1.4) holds, then there is a constant c₃>0such that for each integer n≥1,

ln(X)≤c3n^−a(1 + logn)^b+1. (1.5) Remark 1.3 (Li and Linde 1999)

When E is K -convex (e.g. L^p,1<p<∞) then (1.3) and (1.4) are equivalent.

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The main three goals of our talk are the following:

(a) To connect the H¨older regularity (in the mean-square sense) of a centered continuous real-valued Gaussian process with the rate of convergence of its l-numbers.

(b) To present the Meyer, Sellan and Taqqu wavelet series representations of fractional Brownian motion (fBm), and to show that they are optimal; notice that fBm is the most natural real-valued continuous Gaussian process which extends Brownian motion.

(c) To investigate, for the Riemann-Liouville process (that is the high-frequency part of fBm), the optimality of the series representations obtained via the Haar and the trigonometric bases ofL²[0,1].

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Organization of the talk

1 Introduction

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t∈[0,1]

Gaussian process with continuous paths.

Definition 2.1

Let H∈(0,1]. One says that X is H-H¨older (in the mean-square sense), if there exists a constant c>0 such that for all t,s∈[0,1]^N,

E |X(t)−X(s)|²

≤c|t−s|^2H. (2.1)

Remark 2.1

1 It follows from (2.1) that the paths of X are almost surelyγ-H¨older functions for anyγ∈(0,H); that is, for all t,s∈[0,1]^N,

X(t, ω)−X(s, ω)

≤C⁰(ω)|t−s|^γ. (2.2)

2 Conversely, if the paths of X are almost surely H-H¨older functions then (2.1)

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A classical example of a real-valued, centered, continuous andH-H¨older Gaussian process, is the multivariatefractional Brownian motionBH ={BH(t)}_t∈[0,1]N

whose covariance is given by:

EBH(t)BH(s) = 1

2 |t|^2H+|s|^2H− |t−s|^2H

. (2.3)

The following theorem, is the main result of this section.

Theorem 2.1 (A. and Linde 2008)

Assume that X ={X(t)}_t∈[0,1]N is a real-valued, centered, continuous and H-H¨older Gaussian process. Then there exists a constant c >0 such that for every n≥2,

ln(X)≤cn^−H/Np

logn. (2.4)

For proving Theorem 2.1 we will use the operators theory.

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LetX ={X(t)}_t∈[0,1]N be a centered continuous real-valued Gaussian process.

1 There existsu:L²=L²[0,1]→ C=C[0,1]^N a bounded operator such that for anyt,s∈[0,1]^N,

EX(t)X(s) =hu^∗δt,u^∗δsi_L2, (2.5) whereu^∗ :C^∗→L²is the dual operator ofu, andδt is the Dirac measure in t∈[0,1]^N; one callsu: an operator generating X.

2 In fact, one can prove thatubelongs to the the class

G=G(L²[0,1],C[0,1]^N), that is for each orthonormal basis(f_k)_k≥1 of L² and any sequence{_k}_k_≥1 of independent real-valuedN(0,1) Gaussian random variables the sum

+∞

Xku(fk),

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Some useful properties ofG=G(L²[0,1],C[0,1]^N)

Anyv ∈ Gis a compact operator (see for example the book of Pisier (1989)).

Thel-norm of anyv ∈ Gis defined as

l(v) =



E

+∞

X

k=1

kv(fk)

2

∞





1/2

; (2.6)

this norm is known to be independent of the special choice of the orthonormal basis (fk)k≥1.

(G,l) is a Banach space (Linde and Peitsch 1973).

Ifv has finite rank andX0is an arbitrary standard Gaussian random variable with values in (kerv)^⊥. Then

l(v) = Ekv(X0)k²_∞^1/2 .

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Another useful property ofG=G(L²[0,1],C[0,1]^N) Letv ∈ G and define the kernelK : [0,1]^N×[0,1]→Rby

K(t,x) = (v^∗δt)(x). (2.7)

Thent 7→Φ(t) =K(t,·) is a continuous function from [0,1]^N intoL²[0,1].

Moreover, for anyh∈L²[0,1] andt ∈[0,1]^N one has (vh)(t) =

Z 1 0

K(t,x)h(x)dx. (2.8)

Proof of (2.8): Using (2.7), one obtains that for allt ∈[0,1]^N, Z 1

0

K(t,x)h(x)dx =hv^∗δt,hi_L2,L²=hvh, δtiC,C^∗ = (vh)(t).

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Proof of the continuity ofΦ :t7→K(t,·): Using (2.7) and kfkL² = sup

khk_L2≤1

|hf,hi|

for allf ∈L², one gets that for any t1,t2∈[0,1]^N, kK(t1,·)−K(t2,·)k_L2

=kv^∗(δ_t₁)−v^∗(δ_t₂)k_L2

= sup

khk_L2≤1

|hv^∗(δt₁)−v^∗(δt₂),hi|

= sup

khk_L2≤1

|(vh)(t1)−(vh)(t2)|. (2.9) Sincev is compact, the set{v(h) : khk_L2 ≤1}is equicontinuous (Arzela-Ascoli Theorem).

Thus (2.9) implies that Φ is a continuous function from [0,1]^N intoL²[0,1].

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Proposition 2.1

Let X ={X(t)}_t∈[0,1]N be a centered, continuous and real-valued Gaussian process. Let u∈ G(L²,C)be an operator generating X i.e. for all t,s∈[0,1]^N,

EX(t)X(s) =hu^∗δt,u^∗δsiL², (2.10) and

K(t,x) = (u^∗δ_t)(x), (2.11) the kernel corresponding to u. Then

law

X(t) : t ∈[0,1]^N =law Z 1

0

K(t,x)dB(x) : t ∈[0,1]^N

, where dB is a Brownian measure.

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Proof: It follows from (2.10) and (2.11) that for allt,s∈[0,1]^N, EX(t)X(s) =

Z 1 0

K(t,x)K(s,x)dx.

On the other hand, using the isometry property of Wiener integral one has for all t,s∈[0,1]^N,

E Z 1

0

K(t,x)dB(x) Z 1

0

K(s,x)dB(x)

= Z 1

0

K(t,x)K(s,x)dx.

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Corollary 2.1

Let X ={X(t)}_t∈[0,1]N be a centered, continuous and real-valued Gaussian process. Recall that X can be represented as,

X(t) =

+∞

X

k=1

kαk(t).

In fact, a possible (but not unique) choice of the continuous functionsα_k is αk =u(fk),

where u is an operator generating X and(fk)k≥1is an arbitrary orthonormal basis of L²[0,1].

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Proof: The processX can be represented for everyt ∈[0,1]^N as X(t) =

Z 1 0

K(t,x)dB(x), (2.12)

whereK is the kernel corresponding toui.e. for everyh∈L²andt ∈[0,1]^N, (uh)(t) =

Z 1 0

K(t,x)h(x)dx. (2.13)

Expanding, for all fixedt ∈[0,1]^N, the functionK(t,·) in the basis (fk)k≥1and then using (2.13), one obtains that

K(t,·) =

+∞

X

k=1

(uf_k)(t)f_k, (2.14)

where the series converges inL²([0,1],dx). Finally, it follows from (2.14), (2.12) and the isometry property of Wiener integral that

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X(t) =

+∞

X

k=1

_k(uf_k)(t), (2.15)

where

k = Z 1

0

fk(x)dB(x), for allk.

Remark 2.2

A priori, the series in (2.15) converges in L²(Ω,R)for every fixed t. However, since u∈ G(L²[0,1],C[0,1]^N), one can show that it also converges almost surely uniformly in t∈[0,1]^N or equivalently that it converges in L²(Ω,C[0,1]^N).

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The sequence ofl-numbers of u, is defined as,

ln(u) = inf









E

∞

X

k=n

ku(fk)

2

∞





1/2

: (fk)k≥1ONB inL²[0,1]







; (2.16)

recall that, the sequence ofl-numbers ofX is defined as:

ln(X) = inf









E

∞

X

k=n

kαk

2

∞





1/2

: X =

∞

X

k=0

kαk a.s.







. (2.17)

According to Corollary 2.1, a possible choice of theαk’s isαk =u(fk) for allk;

hence

ln(X)≤ln(u). (2.18)

Observe that, the reverse inequality is also true, thus,

ln(X) =ln(u). (2.19)

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Recall that the goal of this section is to prove that:

Theorem 2.1

Assume thatX ={X(t)}_t∈[0,1]N is a real-valued, centered, continuous and H-H¨older Gaussian process. Then,

ln(X)≤ O

n^−H/Np logn

. (2.20)

Reformulation of Theorem 2.1 in terms of operators theory:

Theorem 2.1 (reformulated)

LetH∈(0,1] andlet u∈ G(L²[0,1],C[0,1]^N) be aH-H¨older operator, which means that, there exists a constantc₂>0 such that for anyh∈L²[0,1] and t₁,t₂∈[0,1]^N,

|(uh)(t )−(uh)(t )| ≤c khk 2|t −t |^H. (2.21)

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To prove Theorem 2.1 (reformulated), we will use the following proposition:

Proposition 2.2

Let u∈ G(L²[0,1],C[0,1]^N). For each a>0 and b∈R, (i)is equivalent to(ii).

(i) The l -numbers of u, satisfies,

ln(u) =O n^−a(logn)^b

. (2.22)

(ii) There exists a sequence(uj)j≥1offinite rank operators, such that:

1 One has, in the sense of the l -norm, u=

∞

X

j=1

uj;

2 the ranks of the uj’s satisfy

rk(uj) =O 2^j

(2.23) and, their l -norms satisfy

_−aj _b

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t_k^j =k/2^j andλ^j_k = [t_k^j,t_k^j₊₁),

with the convention thatλ^j₂_j ={1}. We denote byB([0,1]), the Banach space of the real-valued bounded function on [0,1], equipped with the uniform norm. Let vj :L²[0,1]→B([0,1]) be the finite rank operator defined for allh∈L²as,

vjh=

2^j

X

k=0

(uh)(t_k^j)1_λj k

. (2.25)

It is clear that,

rk (vj)≤2^j+1. (2.26)

moreover, sinceuis a H-H¨older operator, one has ku−vjk=O 2^−jH

. (2.27)

Letu0=v0and for allj≥1,uj =vj−vj−1. (2.26) implies that rk (uj)≤2^j+2 and (2.27) thatu=P+∞

j=0 u_j.It remains to show thatl(u_j) =O 2^−jH√ j+ 1

.

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Organization of the talk

1 Introduction

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covariance,

E(B_H(s)B_H(t)) = 1

2 |s|^2H+|t|^2H− |s−t|^2H

; (3.1)

whenH= 1/2, fBm reduces to Brownian motion, otherwise its increments are correlated and even display long-range dependence whenH∈(1/2,1).

→K¨uhn and Linde (2002), have obtained sharp estimates of the asymptotic behaviour of (l_n(B_H))_n≥1, the sequence of thel-numbers of fBm; namely, there exist two constants 0<c1≤c2, such that for alln≥2,

c1n^−Hp

logn≤ln(BH)≤c2n^−Hp

logn (3.2)

Observe that Theorem 2.1 allows to recover the second inequality in (3.2).

→It seems natural to look for explicit optimal random series representations of fBm; in view of (3.2) their rate of convergence must be,n^−H√

logn.

The main goal of this section is to present the Meyer, Sellan and Taqqu wavelet representations of fBm, and to show that they are optimal.

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Representation of Brownian motion in the Faber-Schauder system

B(t) =0t+

+∞

X

j=0 2^j−1

X

k=0

j,k2^−j/2τ(2^jt−k), (3.3) where:

₀and_j,k are independent real-valued N(0,1) Gaussian random variables, τ is the triangle function based on [0,1],

the series is almost surely uniformly convergent int ∈[0,1].

The expansion (3.3) was introduced by Paul L´evy in 1948. It has turned out to be very useful in the (fine) study of Brownian motion; for instance it allows to prove that Brownian paths do not satisfy a uniform or a pointwise H¨older condition of order 1/2.

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Proposition 3.1

The series expansion of Brownian motion in the Faber-Schauder system is optimal.

For proving Proposition 3.1, we need the following two lemmas.

Lemma 3.1

There is a constant c>0such that for any integer N ≥1 and for each centered Gaussian real-valued sequence Z1, . . . ,ZN one has,

E

sup

1≤k≤N

|Zk|

≤c(1 + logN)^1/2 sup

1≤k≤N E|Zk|²^1/2

. (3.4)

Lemma 3.2

For any t ∈[0,1]and for each integer j≥0, there is at most one integer k, such that0≤k <2^j andτ(2^jt−k)6= 0.

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Proof of Proposition 3.1: Lemma 3.2 implies that for allj≥0 andt ∈[0,1],

2^j−1

X

k=0

|j,k||τ(2^jt−k)| ≤ sup

0≤k<2^j

|j,k|

!

kτk∞. (3.5)

Using (3.5), the gaussianity of the_j,k’s and Lemma 3.1, one has for allm∈N, Qm=E



 sup

t∈[0,1]

B(t)−0t−

m

X

j=0 2^j−1

X

k=0

2^−j/2j,kτ(2^jt−k)





≤c₁

+∞

X

j=m+1

2^−j/2E sup

0≤k<2^j

|j,k|

!

≤c₂

+∞

X

j=m+1

2^−j/2(1 +j)^1/2 sup

0≤k<2^j

E|_j,k|²

!1/2

=c2 +∞

X

j=m+1

2^−j/2(1 +j)^1/2≤c32^−m/2(1 +m)^1/2. {B(t)}

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We suppose that:

ψis a smooth and well-localized real-valued mother wavelet defined onR, {2^j/2ψ(2^jt−k);j∈Zandk∈Z}is an orthonormal basis ofL²(R), ψ, the Fourier transform ofb ψ, is a smooth and well-localized function.

Under these assumptions, for anyH∈(0,1), Ψ_H(t) =

Z

R

e^itξ ψ(ξ)b

(iξ)^H+1/2dξand Ψ_−H(t) = Z

R

e^itξ(iξ)^H+1/2ψ(ξ)b dξ, thefractional primitiveofψof orderH+ 1/2, and itsfractional derivativeof orderH+ 1/2, are well-defined, smooth and well-localized real-valued functions.

Theorem 3.1 (Meyer, Sellan and Taqqu 1999)

The sequences of functions{2^j/2Ψ (2^jt−k);j∈Zand k ∈Z}and

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These bases are well-adapted to the analysis of fBm:

Theorem 3.2 (Meyer, Sellan and Taqqu 1999) The fBm{B_H(t)}_t∈[0,1] can be expressed as:

B_H(t) =

+∞

X

j=−∞

+∞

X

k=−∞

2^−jH Ψ_H(2^jt−k)−Ψ_H(−k)

_j,k, (3.6) where the_j,k’s are independentN(0,1)Gaussian random variables and the series is almost surely uniformly convergent in t. Moreover,

j,k = 2^j(H+1) Z

R

BH(t)Ψ_−H(2^jt−k)dt. (3.7)

→(3.6) is almost a Karhunen-Loeve expansion ofBH.

→It allows to obtain some local and asymptotic properties ofBH (nowhere differentiability, moduli of continuity, behaviour at infinity, and so on).

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Proof of Theorem 3.2: Let us start from theharmonizable representationof fBm:

BH(t) = Z

R

e^itξ−1

(iξ)^H+1/2dB(ξ),b (3.8)

wheredBb is the ”Fourier transform” of the Brownian measuredB. Expanding the functionξ7→ _(iξ)^e^itξ_N+1/2⁻¹ in the orthonormal basis

n2^−j/2e^ikξ/2^jψ(2b ^−jξ);j ∈Zandk ∈Z o

and then, using the isometry property of the integral Z

R

(·)dB, it follows thatb

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B_H(t) = Z

R

e^itξ−1

(iξ)^H+1/2dB(ξ) =b X

j,k∈Z

α_j,k(t)_j,k, (3.9) where the

j,k = 2^−j/2 Z

R

e^ikξ/2^jψ(2b ^−jξ)dB(ξ)b

are independentN(0,1) real-valued Gaussian random variables and αj,k(t) = 2^−j/2

Z

R

e^itξ−1

(iξ)^H+1/2e^−ikξ/2^jψ(2b ^−jξ)dξ. (3.10) The series (3.9) is almost surely uniformly convergent int (Itˆo-Nisio Theorem).

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Let us now prove that

αj,k(t) = 2^−jH ΨH(2^jt−k)−ΨH(−k)

. (3.11)

As the first two moments of the waveletψ vanish, one has,

ψ(ξ) =b O(ξ²), (3.12)

which implies that αj,k(t) = 2^−j/2

Z

R

e^i(t−k^/2^j^)ξ ψ(2b ^−jξ)

(iξ)^H+1/2dξ−2^−j/2 Z

R

e^−ikξ/2^j ψ(2b ^−jξ) (iξ)^H+1/2dξ.

(3.13) Finally, settingη= 2^−jξin the latter two integrals, one obtains (3.11).

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Wavelet representation with a well-localized scaling function

The mother waveletψsatisfies the same conditions as previously andφis a corresponding scaling function.

Asφ(0) = 1b 6= 0, the fractional primitive of φof orderH+ 1/2, can only be defined whenH∈(0,1/2); moreover, it is irregular and bad localized.

⇒The problem of finding a wavelet expansion of fBm with a well-localized scaling function is tricky. However, we need to have this well-localization for the expansion to be optimal.

To overcome this difficulty Meyer, Sellan and Taqqu have replaced the fractional primitive ofφ, by the function ΦH defined as,

ΦbH(ξ) =

1 +e^−iξ iξ

^H+1/2

Φ(ξ).b (3.14)

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LetΨH andΨ_−H be the fractional primitive of order H+ 1/2 and the fractional derivative of order H+ 1/2, of the mother waveletψ. LetΦH be as in (3.14) and letΦ−H be defined as,

Φb−H(ξ) =

1 +e^−iξ iξ

^−H−1/2

Φ(ξ).b (3.15)

Then for any m∈Z, n

2^m/2ΦH(2^mt−l) : l∈Z o [ n

2^j/2ΨH(2^jt−k) : j ∈Z,j≥m,k ∈Z o

and n

2^m/2Φ−H(2^mt−l) : l∈Z o [ n

2^j/2Ψ−H(2^jt−k) : j∈Z,j ≥m,k ∈Z o

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For any m∈Z, the fBm {BH(t)}_t∈[0,1] can be expressed as:

BH(t) = 2^−mH

+∞

X

l=−∞

ΦH(2^mt−l)S_m,l^(H)

+

+∞

X

j=m +∞

X

k=−∞

2^−jHΨ_H(2^jt−k)−b₀, (3.16)

where{S_m,l^(H)}l∈Zis a FARIMA(0,H-1/2,0) random walk and{j,k}j≥m,k∈Z is a sequence of independent real-valuedN(0,1)Gaussian random variables. The series in (3.16), are almost surely uniformly convergent in t; moreover,

S_m,l^(H)= 2^m(H+1) Z

R

BH(t)Φ_−H(2^mt−l)dt and

_j,k = 2^j(H+1) Z

B_H(t)Ψ_−H(2^jt−k)dt.

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The main two advantages of the latter representation with respect to the representation without scaling function are the following.

1 The first term isolates the low frequencies and gives the tendency while the second term involves fluctuations around it.

2 FBm can be approximated by the first term and Mallat pyramidal algorithm allows to compute by induction the coefficientsS_m,l,l∈Zfor any m≥1.

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Optimality of both representations

Theorem 3.5 (A. and Taqqu 2003) For each integer J≥0, let

FJ ={(j,k)∈Z²; 0≤j≤J and|k| ≤(J−j+ 1)⁻²2^J+4},and

P_J ={(j,k)∈Z²; −J ≤j ≤ −1 and|k| ≤2^J/2}.For every integer n≥β with β= 128P+∞

p=1p⁻², letIn⊂Z²be a set satisfying the following properties:

In contains, at most, n indices(j,k),

for all n≥β, F_J(n)∪P_J(n)⊂ I_n, where J(n)is the unique integer such that β2^J(n)≤n< β2^J⁽ⁿ⁾⁺¹.

At last let

BH,n(t) = X

(j,k)∈In

2^−jH ΨH(2^jt−k)−ΨH(−k)

j,k. (3.17) Then there is a r. v. C >0of finite moments such that a.s. for all n≥β,

sup |B (t)−B (t)| ≤Cn^−H(1 + logn)^1/2. (3.18)

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Theorem 3.6 (A. and Taqqu 2003)

We suppose thatβ, FJ and J(n)are the same as in Theorem 3.5. For every integer J≥0let Q_J ={l∈Z;|l| ≤2^J}.For every integer n≥β, letI_n⁰ ⊂Zand I_n⁰⁰⊂Z²be two sets satisfying the following properties:

I_n⁰ contains at most n/2 indices l andI_n⁰⁰ contains at most n/2indices(j,k), for every n≥β, Q_J(n) ⊂ I_n⁰ and F_J(n)⊂ I_n⁰⁰.

At last let

BH,n(t) = X

l∈I_n⁰

ΦH(t−l)S_0,l^(H) (3.19)

+ X

(j,k)∈I_n⁰⁰

2^−jH ΨH(2^jt−k)−ΨH(−k) j,k.

Then there is a r. v. C >0of finite moments such that a.s. for all n≥β, sup |B (t)−B (t)| ≤Cn^−H(1 + logn)^1/2. (3.20)

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Remark 3.1

Another optimal series representation of fBm has been introduced in 2003 by Dzhaparidze and Van Zanten. It has some similarities with the expansion of Brownian motion in the trigonometric system.

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1 Introduction

Some generalities on Riemann-Liouville process

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Some generalities on Riemann-Liouville process

{R^α(t)}_t∈[0,1], the Reimann-Liouville process (RLp) of parameterα >1/2, is defined as the Wiener integral,

R^α(t) = 1 Γ(α)

Z 1 0

(t−s)^α−1₊ dB(s), (4.1) with the convention that for all (x, γ)∈R², (x)^γ₊=x^γ ifx >0 and (x)^γ₊= 0 else.

Remark 4.1

{R¹(t)}_t∈[0,1] is the usual Brownian motion.

If1/2< α <3/2, then{R^α(t)}_t∈[0,1] differs from{B_α−1/2(t)}_t∈[0,1], the fBm of Hurst parameter H=α−1/2, only by a very smooth process, namely,

Q^α(t) = 1 Γ(α)

Z 1 0

n

(t−s)^α−1₊ −(−s)^α−1₊ o dB(s);

sometime{R^α(t)}_t∈[0,1]is called the high-frequency part of fBm and

α

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An advantage of RLp with respect to fBm is thatits parameterαcan take any value strictly greater than 1/2, while the Hurst parameter of fBm is necessarily obliged to belong to (0,1).

The semigroup property of RLp: for each real numberβ >1/2, we denote byC^β−1/2[0,1], the H¨older space on [0,1] of orderβ−1/2, and we denote byIβ:L²[0,1]→ C^β−1/2[0,1] the fractional primitive operator of orderβ, defined, for allh∈L²[0,1] andt ∈[0,1] as,

(I_βh)(t) = 1 Γ(β)

Z 1 0

(t−s)^β−1₊ h(s)ds; (4.2) then one has (IβR^α)(t) =R^α+β(t).

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Series representation of RLP by an orthonormal basis:

LetF= (f_k)_k≥1be an arbitrary orthonormal basis ofL²[0,1]. Expanding, for every fixedt ∈[0,1], the function s7→(t−s)^α−1₊ in this basis, one gets that

(t− ·)^α−1₊ =

∞

X

k=1

(Iαfk)(t)ϕk(·). (4.3) Next, using the isometry property of Wiener integral, it follows that

R^α(t) =

∞

X

k=1

(I_αf_k)(t)_k, (4.4)

where

(k)k≥1=Z 1 0

fk(s)dB(s)

k≥1, (4.5)

is a sequence of independent real-valuedN(0,1) Gaussian random variables.

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The previous series is called the series representation of RLp by the basisF.

A priori, this series is, for every fixed t, convergent in L²(Ω), whereΩdenotes the underlying probability space. However, Itˆo-Nisio Theorem allows to show that it is also convergent, with probability1, uniformly in t.

Theorem 4.1 (K¨uhn and Linde 2002)

For allα >1/2, there are two constants 0<c1≤c2such that for every n≥2, c₁n^−α+1/2p

logn≤l_n(R^α)≤c₂n^−α+1/2p logn, where l_n(R^α)denotes the nth l -number of RLp.

Theorem 4.2 (Schack 2009)

For allα >1/2, any Daubechies wavelet basis of L²[0,+∞)with at least

max{2[α 7}

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Optimality of classical series representations

The main two results of this section:

For all1/2< α <3/2, the Haar basis of L²[0,1],

H:={1} ∪ {hj,k : j ∈N⁰ and0≤k<2^j}, where

hj,k = 2^j/2 1_[ 2k

2j+1,^2k+1

2j+1]−1_[2k+1 2j+1,^2k+2

2j+1]

,

generates an optimal series representation of R^α. Theorem 4.4 (A. and Linde 2009)

For all1< α≤2, the trigonometric basis of L²[0,1], T:={1} ∪ {√

2 cos(kπ·) : k ∈N},

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Remark 4.3

Whenα >3/2, the random series representation of R^αgenerated by the Haar basis is rearrangement non-optimal i.e. the optimality does not hold even if the terms of the series are renumbered (A. and Linde 2009); the latter result remains true whenα= 1(Lifshits 2009).

The random series representation of R^α generated by the trigonometric basis is rearrangement non-optimal ifα >2(A. and Linde 2009).

K¨uhn and Linde (2002) have shown that the random series representation of R^αgenerated by the trigonometric basis is optimal ifα∈(1/2,1].

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From now on, we will mainly focus on the proof of the theorem which concerns the Haar basis.

Let us set

h−1=1and−1= Z 1

0

h−1(s)dB(s).

We denote by{j,k : j∈N0,0≤k≤2^j−1}, the sequence of the real-valued N(0,1) Gaussian random variables, defined as,

j,k = Z 1

0

hj,k(s)dB(s).

At last, recall thatI_α:L²[0,1]→C^α−1/2[0,1], is the fractional primitive operator of orderα.

The series representation ofR^α by the Haar basis can be expressed as:

R^α(t) =−1(Iαh−1)(t) +

+∞

X

j=0 2^j−1

X

k=0

j,k(Iαhj,k)(t). (4.6)

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R_J^α(t) =₋₁(I_αh₋₁)(t) +

J−1

X

j=0 2^j−1

X

k=0

_j,k(I_αh_j,k)(t), (4.7) in which there are exactly 2^J terms.

Suppose1/2< α <1. Then there is a random variable C >0of finite moments of any order, such that one has almost surely, for every J∈N,

sup

t∈[0,1]

R^α(t)−R_J^α(t)

≤C2^{−(α−1/2)J}√ 1 +J.

Suppose1< α <3/2. Then there is a constant c>0, such that one has almost surely, for every J ∈N,

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In order to be able to prove the latter two theorems, we need some preliminary results.

Remark 4.4 One has

(Iαhj,k)(t)

= 2^j/2 Γ(α+ 1)

(Z (2k+1)/2^j+1 2k/2^j+1

(t−s)^α−1₊ ds−

Z (2k+2)/2^j+1 (2k+1)/2^j+1

(t−s)^α−1₊ ds )

= 2^j/2 Γ(α+ 1)

t−2k+ 2 2^j+1

^α

+−2

t−2k+ 1 2^j+1

^α

+

+ t− 2k

2^j+1 ^α

+

.

The following lemma allows to estimate (Iαhj,k)(t).

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For each realsγ >0 and t∈[0,1], and for all integers j ∈N0and0≤k≤2^j−1, we set

Aγ,j,k(t) :=

t−2k+ 2 2^j+1

^γ

+−2

t−2k+ 1 2^j+1

^γ

++ t− 2k

2^j+1 ^γ

+. (4.8) Furthermore, letekj(t)the unique integer satisfying:

ekj(t)

2^j ≤t <ekj(t) + 1

2^j . (4.9)

with the convention thatke_j(1) = 2^j−1. Then, (i) for all k ≥ek_j(t) + 1, one has A_γ,j,k(t) = 0;

(ii) there is a constant c>0, only depending on γ, such that for all j∈N0and 0≤k ≤ekj(t), one has

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The following lemma, results from Borel-Cantelli Lemma; it provides sharp estimates of the asymptotic behaviour of the sequence of the real-valuedN(0,1) Gaussian random variables,{j,k : j∈N0,0≤k ≤2^j−1}.

Lemma 4.2

There exists a random variable C >0of finite moment of any order such that one has almost surely, for every j ∈N0and0≤k <2^j−1,

|j,k| ≤Cp 1 +j.

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It follows from the latter two lemmas that almost surely for everyt ∈[0,1] and J∈N,

|R^α(t)−R_J^α(t)| ≤

+∞

X

j=J 2^j−1

X

k=0

|j,k||(Iαh_j,k)(t)|

≤ C1 +∞

X

j=J

2^{−j(α−1/2)}p j+ 1

ekj(t)

X

k=0

1 +ekj(t)−kα−2

≤ C22^−J^(α−1/2)√ J+ 1.

Observe that the condition 1/2≤α <1 plays a crucial role in the proof. Indeed, one has

ekj(t)

X

k=0

1 +ekj(t)−kα−2

≤

+∞

X

p=1

p^α−2<∞, (4.11)

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Main ideas of the proof of the optimality when1< α <3/2:

We want to show that there is a constantc₁>0, such that one has for every J∈N,

E(A_J)≤c₁2^{−(α−1/2)J}√

1 +J, (4.12)

where

AJ = sup

t∈[0,1]

|R^α(t)−R_J^α(t)|. (4.13)

First step: We prove that there is a constantc2>0, such that one has for every J∈N,

E(BJ)≤c22^{−(α−1/2)J}√

1 +J, (4.14)

where

BJ = sup

0≤K<2^J,K∈N0

|R^α(K2^−J))−R_J^α(K2^−J))|. (4.15)

Second step: We prove that there is a constantc₃>0, such that one has for everyJ ∈N,

0≤ (A −B )≤c 2^{−(α−1/2)J}√

1 +J. (4.16)

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Lemma 4.3

There exists a constant c >0 such that for any N∈Nand any real-valued centered Gaussian sequence Z₁, . . . ,Z_N one has,

E n

sup

1≤k≤N

|Zk|o

≤c(1 + logN)^1/2 sup

1≤k≤N

E|Zk|²^1/2

. (4.17)

Therefore, there is a constantc4>0 such that for allJ∈N, E sup

0≤K<2^J,K∈N0

R^α(K2^−J))−R_J^α(K2^−J))

≤c₄(1 +J)^1/2 sup

0≤K<2^J,K∈N0

E

R^α(K2^−J))−R_J^α(K2^−J))

21/2

.

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Lemma 4.4

There is a constant c>0such that one has for all t ∈[0,1]and J∈N, E

R^α(t)−R_J^α(t))

2

≤c2^{−J(2α−1)}.

Proof Lemma 4.4: Using the fact that{j,k : j∈N0,0≤k ≤2^j−1} is a sequence of independent real-valuedN(0,1) Gaussian random variables and using the previous estimations of the (Iαhj,k)(t)’s, one gets that for allJ ∈N,

E

R^α(t)−R_J^α(t))

2

=E X^+∞

j=J 2^j−1

X

k=0

_j,k(I_αh_j,k)(t)²

=

+∞

X

j=J 2^j−1

X

k=0

|(Iαh_j,k)(t)|²

≤c1 +∞

X

j=J

2^{−j(2α−1)}

ekj(t)

X

k=0

1 +ekj(t)−k−2(2−α)

≤c22^{−J(2α−1)}.

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Optimality of the representention by the trigonometric basis:

Roughly speaking, whenα∈(1/2,2], the optimality of the representention by the trigonometric basis, can be shown, by following the main lines as in the proof of the latter theorem, and by using the fact there is a constantc>0 such that for all integerk≥1,

sup

t∈[0,1]

|(Iα{cos(kπ·)})(t)| ≤ck^−α (4.18) and

sup

t∈[0,1]

|(Iα{sin(kπ·)})(t)| ≤ck^−α (4.19)