Proofs

Theorem 5.6.1 is an extension of [Arcones, 1994, Theorem 4] to arrays of stationary Gaussian processes in the unidimensional case and Theorem 5.6.2 extends the result of Cs¨org´o and Mielniczuk [1996] to arrays of stationary Gaussian processes. These two theorems are useful for the proof of Proposition 5.3.1.

Theorem 5.6.1. Let {Xj,i, j ≥ 1, i ≥ 0} be an array of standard stationary Gaussian

where g_∞ is a 2π-periodic function which is bounded on (−π, π) and continuous at the origin. Let h be a function on R with Hermite rank τ ≥ 1. We assume that h is ei-ther bounded or is a finite linear combination of Hermite polynomials. Let {n_j}j≥1, be a sequence of integers such that n_j tends to infinity as j tends to infinity. Then,

√1n_j

In the previous equality, c_ℓ = E[h(X)H_ℓ(X)], where H_ℓ is the ℓ-th Hermite polynomial and X is a standard Gaussian random variable.

Proof of Theorem 5.6.1. Let us first prove that Pnj Using Mehler’s formula, see Eq. (2.1) of Breuer and Major [1983], we have

Var

By [Taqqu, 1977, Lemma 3.2 P. 210],E[H_l1(X_j,i1), . . . , H_lm(X_j,im)] is zero ifl₁+· · ·+l_m is odd. Otherwise it is bounded by a constant times a sum of products of (l1+· · ·+lm)/2 correlations. Bounding, in each product, all of them butp+ 1, byρ^∗ <1/(2p), we get that E

H_l1(X_j,i1), . . . , H_l2p+1(X_j,i2p+1)

is bounded by a finite number of terms of the following form

(ρ^∗)

l1+···+l2p+1

2 −(p+1)ρ_j(i₂−i₁)ρ_j(i₄−i₃). . . ρ_j(i_2p−i_2p−1)ρ_j(i_2p+1−i_2p)E H_l1(X). . . H_l2p+1(X) , whereX is a standard Gaussian random variable. Note also that the hypercontractivity

[Taqqu, 1977, Lemma 3.1 P.210] yields E

H_l1(X). . . H_l2p+1(X)≤(2p)

l1+···+l2p+1

2 p

l₁!. . . l_2p+1!.

Thus, using the Cauchy-Schwarz inequality and that ρ^∗ < _2p¹, there exists a positive constant C such that To conclude the proof of (5.31), it remains to prove that

P Let us first study the numerator in the l.h.s of (5.34).

X To prove (5.34), we start by proving that

nj

Using the notationD_nj(λ) =Pnj Using (5.28), the boundedness of g_∞ and that u_j is bounded, there exists a positive constant C such that The result (5.35) thus follows from the convergence of the integral in (5.36) which is proved in Lemma 5.7.3. Let us now prove that

1 as n tends to infinity. The second term in the r.h.s of (5.38) can be upper bounded as follows. For 0< η≤π, the first and last terms in the r.h.s of (5.39) are bounded byCπ/(n_jη²). The continuity of g_∞ at 0 and the fact thatRη

−ηF_j(λ)dλ≤Rπ

−πF_j(λ)dλ= 1 ensure that the second term in the r.h.s of (5.39) tends to zero asn tends to infinity. This concludes the proof of (5.37).

Using the same arguments as those used to prove (5.37) and the fact that ρ^l_j is the autocorrelation associated tof_j^⋆l which is the l-th self-convolution offj, we get that

1 applying Lemma 5.7.6 withfn, gn,f and g defined hereafter.

f_nj(s, l) = c²_l

Using (5.28) and the fact that the spectral density associated to ρ^l_j is f_j^⋆l, we get, as n→ ∞, Using [Moulines et al., 2007b, Lemma 1], we get

X Hence we get (5.34) by noticing that the numerator in (5.34) isO(n^p_j).

If Condition (5.33) is not satisfied then let k0 be such that ρj(k) ≤ ρ^∗ <1/(2p), for all k > k₀. In the case where h is a linear combination of L Hermite polynomials, the same arguments as those used previously are valid with ρ^∗ = 1. In the case where h is bounded, there exists a positive constantC such that

E

By expanding|h| onto the basis of Hermite polynomials, we can conclude with the same arguments as those used when Condition (5.33) is valid.

b) Let us now assume that|{i₁, . . . , i_2p+1}|=r≤2p. In the case wherehis bounded, the inequality (5.41) is valid with q ≤ r which gives that the numerator of (5.31) is

O(n^⌊_j^r/2⌋).In the case where h is a linear combination of L Hermite polynomials, we use the same arguments as those used in a) withρ^∗ = 1 which implies that the numerator of (5.31) isO(n^⌊_j^r/2⌋). By [Rosenblatt, 1985, Formula (33), P.69], we have

E

indicates that we are to sum over all symmetric matrices ν with nonnegative integer entries, ν_ii= 0 and the row sums equal tol₁, . . . , l_2p.

We shall prove that among all the terms in the r.h.s of (5.43), the leading ones corre-spond to the case where we havep pairs of equal indices in the set {l₁, . . . , l_2p}, that is, for instance,l₁ =l₂, l₃ =l₄, . . . , l_2p−1=l_2p and ν_1,2 =l₁,ν_3,4 =l₃,...,ν_2p−1,2p =l_2p−1 the othersν_i,j being equal to zero. This gives

(l₂!)². . .(l_2p!)²ρ_j(i₂−i₁)^l2ρ_j(i₄−i₃)^l4. . . ρ_j(i_2p−i_2p−1)^l2p l₂!. . . l_2p! . The corresponding term in (5.42) is given by

X which corresponds to the denominator in the l.h.s of (5.32). Since there exists exactly (2p)!/(2^pp!) possibilities to have pairs of equal indices among 2p indices we obtain (5.32) if we prove that the other terms can be neglected.

Let us first consider the case where

∀i, ρj(i)≤ρ^∗ < 1

2p−1 (5.44)

and|{i₁, . . . , i_2p}|= 2p. By [Taqqu, 1977, Lemma 3.2 P. 210],E[H_l1(X_j,i1), . . . , H_lm(X_j,im)]

is zero ifl₁+· · ·+l_mis odd. Otherwise it is bounded by a constant times a sum of products of (l₁+· · ·+l_m)/2 correlations. Bounding, in each product, all of them but p+ 1, by

ρ^∗ <1/(2p−1), we get thatE

H_l1(X_j,i1), . . . , H_l2p(X_j,i2p)

is bounded by a finite number of terms of the following form

(ρ^∗)

l1+···+l2p

2 −(p+1)ρ_j(i₂−i₁)ρ_j(i₄−i₃). . . ρ_j(i_2p−i_2p−1)ρ_j(i_2p−i₁)E H_l1(X). . . H_l2p(X) . where X is a standard Gaussian random variable. Using the same arguments as in the case wherem was odd, we have

X To have the result (5.32), it remains to show that

P1≤i1,...,i2p≤nj The numerator of (5.45) can be rewritten as

X

Using (5.37), we have P

1≤i36=i4≤njρj(i4−i3)p−2 Using (5.28) and thatg_∞is bounded, (5.46) will follow if we prove thatRπ

−π(Rπ

−π|D_nj(λ)||D_nj(λ+

ξ)|dλ)²dξ=O(n_j).Since there exists a positive constantcsuch that|D_nj(λ)| ≤cn_j/(1 +

n_j|λ|), for allλin [−π, π], The result (5.46) thus follows from the convergence of the last integral in (5.47) which is proved in Lemma 5.7.4. Hence we get (5.45) since the numerator of the l.h.s of (5.45) is O(n^p_j⁻¹) and the denominator is O(n^p_j) by the same arguments as those used to find the order of the denominator of (5.34). If Condition (5.44) is not satisfied or if|{i1, . . . , i2p}|<

2p, we can use similar arguments as those used in 1)a) and 1)b) to conclude the proof.

Theorem 5.6.2. Let {X_j,i, j ≥ 1, i ≥ 0} be an array of standard stationary Gaussian

wherec_l is the l-th Hermite coefficient of the function h defined by h(·) = the CLT (5.49) follows from Theorem 5.6.1.

Let us now prove that there exists a positive constant C and β >1 such that for all r≤s≤t,

E |S_j(s)−S_j(r)|²|S_j(t)−S_j(s)|²

≤C|t−r|^β . (5.50) The convergence (5.48) then follows from (5.49), (5.50) and [Billingsley, 1999, Theorem 13.5]. Note that

wheref_t(X) =¹_{X≤t}−E(¹_{X≤t}).By developing each difference of functions in Hermite Using the same arguments as in the case wherem is even in the proof of Theorem 5.6.1, we obtain variable. Since, by (5.40),Pnj

i,i^′,l,l^′=1ρ^p_j¹(l−i)ρ^p_j²(l^′−i^′) =O(n²_j), we get with the Cauchy-Schwarz inequality that there exists a positive constant C such that

E |S_j(s)−S_j(r)|²|S_j(t)−S_j(s)|² con-cludes the proof of (5.50).

Proof of Proposition 5.3.1. We first prove (5.16) for ∗= CL.

√nj(bσ_CL,j² −σ_j²) = 1

the space of cadlag functions equipped with the topology of uniform convergence. This convergence follows by applying Theorem 5.6.2 toXj,i=Wj,i/σj which is an array of zero mean stationary Gaussian processes by [Moulines et al., 2007b, Corollary 1]. The spectral densityf_j of (X_j,i)_i≥0is given byf_j(λ) =D_j,0(λ;f)/σ²_j whereD_j,0(·;f) is the within scale spectral density of the process{W_j,k}k≥0 defined in (5.8) andσ²_j is the wavelet spectrum defined in (5.9). Here, g_∞(λ) = D_∞,0(λ;d)/K(d), with D_∞,0(·;d) defined in (2.84) and K(d) defined in (2.90) since by [Moulines et al., 2007b, (26) and (29) in Theorem 1]

D_j,0(λ;f)

f^∗(0)K(d)2^2dj −D_∞,0(λ;d) K(d)

≤C LK(d)⁻¹2^−βj →0, asn→ ∞,

σ²_j

f^∗(0)K(d)2^2dj −1

≤C L2^−βj→0, asn→ ∞.

Note also that, by [Moulines et al., 2007b, Theorem 1], g_∞(λ) is a continuous and 2π-periodic function on (−π, π). Moreover, g_∞(λ) is bounded on (−π, π) by Lemma 5.7.5 and

u_j =C₁ 2^−βj

σ_j²/2^2dj 2^−βj+C₂ σ²_j 2^2dj

!

→0, asn→ ∞,

whereC₁ andC₂ are positive constants. The asymptotic expansion (5.16) forbσ_MAD,j can be deduced from the functional Delta method stated e.g in [van der Vaart, 1998, Theo-rem 20.8] and the classical Delta Method statede.g in [van der Vaart, 1998, Theorem 3.1].

To show this, we have to prove that T0 = T1 ◦T2 is Hadamard differentiable and that the corresponding Hadamard differential is defined and continuous on the whole space of cadlag functions. We prove first the Hadamard differentiability of the functional T₁. Let (g_t) be a sequence of cadlag functions with bounded variations such that kg_t−gk∞→0, ast→0,where gis a cadlag function. For any non negative r, we consider

T₁(F_j+tg_t)[r]−T₁(F_j)[r]

t = (F_j+tg_t)(r)−(F_j+tg_t)(−r)−F_j(r) +F_j(−r) t

= tg_t(r)−tg_t(−r)

t =g_t(r)−g_t(−r)→g(r)−g(−r), sincekg_t−gk∞→0, ast→0.The Hadamard differential of T₁ at g is given by :

(DT₁(F_j).g)(r) =g(r)−g(−r).

By [van der Vaart, 1998, Lemma 21.3],T₂ is Hadamard differentiable. Finally, using the Chain rule [van der Vaart, 1998, Theorem 20.9], we obtain the Hadamard differentiability ofT₀ with the following Hadamard differential :

DT₀(F_j).g =−(DT₁(F_j).g)(T₀(F_j))

(T1(Fj))^′[T0(Fj)] =−g(T₀(F_j))−g(−T₀(F_j)) (T1(Fj))^′[T0(Fj)] .

In view of the last expression,DT₀(F_j) is a continuous function ofgand is defined on the whole space of cadlag functions. Thus by [van der Vaart, 1998, Theorem 20.8], we obtain :

and the expansion (5.16) for ∗ = MAD follows from the classical Delta method applied withf(x) =x². We end the proof of Proposition 5.3.1 by proving the asymptotic expan-sion (5.16) for∗= CR. We use the same arguments as those used previously. In this case the Hadamard differentiability comes from [L´evy-Leduc et al., 2009, Lemma 1].

The following theorem is an extension of [Arcones, 1994, Theorem 4] to arrays of stationary Gaussian processes in the multidimensional case.

Theorem 5.6.3. LetX_J,i= origin. Let h be a function on R with Hermite rank τ ≥1 which is either bounded or is a finite linear combination of Hermite polynomials. Let β = {β₀, . . . , β_d} in R^d+1 and H:R^d+1 → Rthe real valued function defined by H(x) =Pd

j=0β_jh(x_j). Let {n_J}J≥1 be a sequence of integers such that nJ tends to infinity asJ tends to infinity. Then

√1n_J

In the previous equality, c_ℓ = E[h(X)H_ℓ(X)], where H_ℓ is the ℓ-th Hermite polynomial and X is a standard Gaussian random variable.

The proof of Theorem 5.6.3 follows the same lines as the one of Theorem 5.6.1 and is thus omitted.

Proof of Theorem 5.3.1. Without loss of generality, we set f^∗(0) = 1. In order to prove (5.20), let us first prove that for α= (α₀, . . . , α_ℓ) where theα_i’s are in R, Thus, proving (5.53) amounts to proving that

2^−ℓ/2f^∗(0)K(d) [Moulines et al., 2007b, (29) in Theorem 1]. Note that

nJ0+j−1

and

is a 2^ℓ+1−1 stationary Gaussian vector. By Lemma 5.7.1, F is of Hermite rank larger than 2. Hence, from Theorem 5.6.3 applied toH(·) =F(·),X_J,i=Y_J0,ℓ,i andh(·) = IF(·),

. By [Moulines et al., 2007b, (26) and (29)] and by using the same arguments as those used in the proof of Proposition 5.3.1, Condition (5.51) of Theorem 5.6.3 holds withf_J^(j,j′)(λ) =D^(r)_J0+j,j−j′(λ;f)/σ_J0+jσ_J0+j^′ and g_∞^(j,j′) = D^(r)_∞,j−j′(λ;d)/K(d), where 0 ≤ r ≤ 2^j−j′ −1 and D_J0+j,j−j^′(·;f) is the cross-spectral density of the stationary between scale process defined in (5.8). Lemma 5.7.5 and [Moulines et al., 2007b, Theorem 1] ensure that D^(r)_∞,j−j′(·;d) is a bounded, continuous and 2π-periodic function.

By using Mehler’s formula [Breuer and Major, 1983, Eq. (2.1)] and the expansion of IF onto the Hermite polymials basis given by: IF(x,∗,Φ) =P Without loss of generality, we shall assume in the sequel that j ≥ j^′. (5.57) can be rewritten as follows by using thati^′= 2^j−j′q+r, whereq∈Nand r∈ {0,1, . . . ,2^j−j′−1}

and Eq. (18) in Moulines et al. [2007b] whereD_J0+j,j−j^′(·;f) is the cross-spectral density of the stationary between scale process

defined in (5.8). We aim at applying Lemma 5.7.6 withf_n,g_n,f andg defined hereafter.

fnJ0+j(τ, p) = c²_p(IF_∗) Using [Moulines et al., 2007b, (26) and (29) in Theorem 1] we get that

nlim→∞ Then, with Lemma 5.7.6, we obtain

e

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