Spectral analysis ofmultifractional LRD functional time series

Fractional Calculus and Applied Analysis https://doi.org/10.1007/s13540-022-00053-z ORIGINAL PAPER Spectral analysis of multifractional LRD functional time series M. Dolores Ruiz-Medina1 Received: 12 October 2021 / Revised: 25 April 2022 / Accepted: 2 May 2022 © The Author(s) 2022 Abstract Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak-consistent parametric estimator of the long-memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss. The results derived allow, in particular, to develop inference from the discrete sampling of the Gaussian solution to fractional and multifractional pseudodifferential models introduced in Anh et al. (Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434– 1459, 2016) and Kelbert (Adv Appl Probab 37(1):1–25, 2005). Keywords Minimum contrast parameter estimation ·Multifractional functional ARIMA models ·Multifractional in time evolution equations ·Spatial-varying long-range dependence range Mathematics Subject Classification 60G10 ·60G12 ·60G18 ·60G20 ·60G22 (primary) ·60G60 1 Introduction One can find evidence of Long Range Dependence (LRD) in time series data arising in several areas like agriculture, environment, economics, finance, geophysics, just to mention a few. Indeed, a huge amount of literature on this topic has been developed over the last few decades (cf., [6,8,18,30,35]). This framework allows the description of BM. Dolores Ruiz-Medina [email protected] 1Department of Statistics and Operation Research, Faculty of Sciences, University of Granada, Campus Fuente Nueva s/n., 18071 Granada, Spain 123 M. D. Ruiz-Medina processes with long persistence in time. In the stationary case, LRD is characterized by a slow decay of the covariance function, and an unbounded spectral density, typically at zero frequency. In the real-valued process framework, we refer to the reader to the papers [1,4,16,17,19,21,25,36], among others. Special attention has been paid to the self-similar asymptotic behavior of the second-order moments of the Gaussian solution to fractional and multifractional linear pseudodifferential equations (see, e.g., [2,3,22]). Particularly, the analysis of LRD phenomena in an infinite-dimensional framework is a challenging topic where several problems remain open. Only a few contributions can be found on this topic in functional processes. That is the case of time-varying isotropic vector random fields on the sphere introduced in [28], that were also analyzed by [27] in the framework of compact two-points homogeneous spaces. On the other hand, LRD in functional sequences is characterized by the nonsummability in time of the nuclear norms of the associated family of covariance operators. In the linear case, a variable-order fractional power law usually characterizes the asymptotic behavior in time of the norms of the functional parameters, given by bounded linear operators. That is the case of the approaches in the current literature based on operator-valued processes. A fractional Brownian motion with values in a Hilbert space, involving an operator-valued Hurst coefficient, is considered in [33](seealso[32] on the functional analytical tools applied). In [14], a central and functional central limit theorems are obtained under non-summability of the operator norm sequence. The limit process in this functional central limit result is a self-similar process, characterized by an operator defining the self-similarity exponent. Note that the LRD models introduced in these papers in the linear setting are characterized and analyzed in the time domain. Recently, in [15], for LRD linear processes in a separable Hilbert space, a stochastic-integral based approach is adopted to representing the limiting process of the sample autocovariance operator in the space of Hilbert–Schmidt operators. A semiparametric linear framework has been adopted to analyze LRD in functional sequences in [26]. The functional dependence structure is specified via the projections of the curve process onto different sub-spaces, spanned by the eigenvectors of the long-run covariance function. A Central Limit Theorem is derived under suitable regularity conditions. Functional Principal Component Analysis is applied in the consistent estimation of the orthonormal functions spanning the dominant subspace, where the projected curve process displays the largest dependence range. The memory parameter and the dimension of the dominant subspace are estimated as well. The conditions assumed are satisfied, in particular, by a functional version of fractionally integrated autoregressive moving averages processes. Some interesting applications to US stock prices and age specific fertility rates are also provided. As follows from the above cited references, the spectral domain has not been exploited yet in the formulation and estimation of LRD in stationary functional time series. Furthermore, LRD functional time series models have mainly been introduced in the linear setting. Our paper attempts to cover these gaps. To this aim, the spectral representation of a self-adjoint operator on a separable Hilbert space, in terms of a spectral family of projection operators, is considered. Suitable conditions are then assumed on the symbol defining such a representation, for the spectral density operator 123 Spectral analysis of multifractional LRD functional time series family at a neighborhood of zero frequency. Specifically, the behavior of the spectral density operator at zero frequency is characterized by a bounded symmetric positive operator family, whose operator norm slowly varies at zero frequency, composed with an unbounded operator at zero frequency involving the long-memory operator. The corresponding covariance operator family displays a heavy tail behavior in time as proved in Proposition 1. As an interesting special case, we refer to a family of fractionally integrated functional autoregressive moving averages processes of variable order (see also Remark 9 in [26]). Several additional examples can be found by tapering, in the frequency domain, the symbols of the spectral density operator family, associated with infinite-dimensional stationary LRD processes in continuous time. Particularly, we consider the case of fractional integration of variable order of functional processes with rational spectral density operator (see, e.g., [2,3,22]). The convergence to zero, in the Hilbert–Schmidt operator norm, of the integrated periodogram bias operator is derived, under the square integrability in the frequency domain of the Hilbert– Schmidt operator norm of the spectral density operator family. This condition holds under mild conditions, in our case, under the second-order property of the functional process, assuming the integrability in the frequency domain of the operator norm of the spectral density operator family. The weak consistency of the proposed parametric estimator of the long-memory operator then follows in the Gaussian case, extending Theorem 3 in [4]. Note that the parametric estimation approach in the spectral domain has not been exploited yet in the functional time series context. Under short-range dependence (SRD), [31] adopts a nonparametric framework. Specifically, a weighted average of the functional values of the periodogram operator is considered as an estimator of the spectral density operator. This methodology is not applicable when one wants to approximate the behavior of the spectral density operator at zero frequency in the presence of LRD. In this paper, we consider a parametric estimator of the long-memory operator, computed by minimizing the operator norm of a weighted Kullback–Leibler divergence operator. This operator compares the behavior at a neighborhood of zero frequency of the true spectral density operator, underlying to the curve data, with the possible semiparametric candidates. On the other hand, this functional is linear with respect to the periodogram operator. This is an important advantage of the proposed estimation methodology in relation to nonparametric kernel estimation. The outline of the paper is the following. Preliminary definitions, results and first conditions are established in Sect. 2. The main assumptions are formulated in Sect. 3. Under this setting of conditions, LRD is characterized in the functional spectral domain. The heavy tail behavior in time of the associated covariance operator family is obtained in Proposition 1. Some examples are provided as well. In Sect. 4, the convergence to zero of the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved in Theorem 1. Under a Gaussian scenario, Theorem 2in Sect. 5derives the consistent parametric estimation of the long-memory operator in the functional spectral domain. Some final comments are given in Sect. 6. 123 M. D. Ruiz-Medina 2 Preliminaries In what follows, (, A,P)denotes the basic probability space. Let Hbe a real separable Hilbert space with the inner product ·,·H,and  H=H+iH,its complex version, whose elements are functions of the form ψ=ϕ1+iϕ2,ϕ i∈H,i=1,2. Its inner product is given by ϕ1+iϕ2,φ 1+iφ2 H =ϕ1,φ 1H−ϕ2,φ 2H+iϕ2,φ 1H +ϕ1,φ 2H.(2.1) Recall that L2  H(, A,P)denotes the space of second-order zero-mean  H-valued random variables on (, A,P), with the norm X2 L2  H(,A,P)=E[X2  H],for every X∈L2  H(, A,P). In the following, fix an orthonormal basis {ϕk,k≥1}of H,and consider {ψk=(1/2)[ϕk+iϕk],k≥1},(2.2) as an orthonormal basis of  H. All the subsequent identities involving operator norms can be expressed in terms of such an orthonormal basis, allowing the interpretation of Has a closed subspace of  H. Particularly, the nuclear ·L1( H),and the Hilbert– Schmidt ·S( H)operator norms on  Hare defined as follows: AL1( H)= k≥1AA1/2(ψk), ψk H, AS( H)=⎡ ⎣ k≥1AA(ψk), ψk H⎤ ⎦ 1/2 =AAL1( H),(2.3) with {ψk,k≥1}being an orthonormal basis of  Has given in (2.2). We denote by · L( H)the norm in the space of bounded linear operators on  H,i.e., AL( H)=supψ∈ H;ψ=1A(ψ) H.This norm is also usually referred as the operator norm (or uniform operator norm). Through the paper we consider the equality between operators on  H(respectively, on H) in the norm of the space L( H) (respectively, of the space L(H)) implying the pointwise identity of such operators over the functions on  H(respectively on H). Otherwise, the norm with respect to which the identity considered holds is established. 123 Spectral analysis of multifractional LRD functional time series For simplicity of notation, in the subsequent development, the letter Kwill refer to a positive constant whose specific value may vary from one to another inequality or identity. Let {Xt,t∈Z}be a strictly stationary functional time series with zero mean E[Xt]=0,and functional variance σ2 X=E[Xt2 H]=E[X02 H]=R0L1(H), for every t∈Z.Also, Rt=E[Xs+t⊗Xs]=E[Xt⊗X0],∀t,s∈Z, Rt(g)(h)=E[Xs+t(h)Xs(g)]=EXs+t,hHXs,gH,∀h,g∈H. (2.4) Note that, E[X02 H]<∞implies P[Xt∈H]=1,for all t∈Z. Let Fωbe the spectral density operator on  H,defined by the following identity in the L( H)norm, for ω∈[−π,π]\{0}: Fω= L( H) 1 2π t∈Z exp (−iωt)Rt.(2.5) Remark 1 In [31], convergence of series (2.5) holds in the nuclear norm for SRD functional sequences. Here, a weaker convergence is assumed. Indeed, identity (2.5) could hold for ω∈[−π,π]\0,where 0dω=0.In our case, 0={0}for the characterization of LRD in Assumption II below. For simplicity, in the following, we will omit the reference to the set [−π, π]\0, when the identities hold almost surely in the frequency domain. That is the case of the identities for a spectral density operator family involving an unbounded spectral density operator at zero-frequency (see Eq. (3.1) below). The functional Discrete Fourier Transform (fDFT)  X(T)of the functional data {Xt,t=1,...,T}is defined as  X(T) ω(·)=  H 1 √2πT T  t=1 Xt(·)exp (−iωt),ω∈[−π,π],(2.6) where =  H denotes the equality in  Hnorm. Hence,  X(T) ωis 2π-periodic and Hermitian with respect to ω∈[−π,π]. Remark 2 Under the condition E[X02  H]<∞,applying triangle inequality, E X(T) ω H≤1 √2πT T  t=1 EXt(·) H<∞, for every ω∈[−π,π].The fDFT  X(T) ωdefines a random element in  H,and P X(T) ω(·)∈ H=1.Hence, F(T) ω=E X(T) ω⊗ X(T) ω∈L1( H), for ω∈[−π,π]. 123 M. D. Ruiz-Medina The periodogram operator p(T) ω= X(T) ω⊗ X(T) ωis an empirical operator, with mean E[p(T) ω]=E[ X(T) ω⊗ X(T) −ω]=F(T) ω.Particularly, under (2.5), for any T≥2,the following identity holds in L( H): F(T) ω=Ep(T) ω=1 2πTT  t=1 T  s=1 exp (−iω(t−s))E[Xt⊗Xs] =1 2π T−1  u=−(T−1) exp (−iωu)(T−|u|) TRu.(2.7) Let FTbe the Féjer kernel, given by FT(ω) =1 T T  t=1 T  s=1 exp (−i(t−s)ω),ω∈[−π,π],T≥2.(2.8) Applying the Fourier Transform Inversion Formula in the space L( H), from Eqs. (2.7) and (2.8), for each ω∈[−π,π], F(T) ω=[FT∗F•](ω) =π −π FT(ω −ξ)Fξdξ, T≥2.(2.9) 2.1 Preliminaries on spectral analysis of self-adjoint operators This section presents some preliminary elements on spectral theory of self-adjoint operators on a separable Hilbert space (see, e.g., [12], pp. 112–140). It is well-known that, for a self-adjoint operator Don a separable Hilbert space  H, there exists a family of projection operators {Eλ,λ∈⊆R},also called the spectral family of D,such that the following identity holds: D= λdEλ.(2.10) This family of projection operators satisfies the following properties: (i) EλEμ=Einf{λ,μ}; (ii) lim λ→λ; λ≥λE λ=Eλ; (iii) limλ→−∞ Eλ=0;limλ→∞ Eλ=I H,where I Hdenotes the identity operator on  H. (iv) The domain of Dis defined as Dom(D)=h∈ H:|λ|2dEλ(h), h H<∞.(2.11) 123 Spectral analysis of multifractional LRD functional time series (v) A continuous function G(D)admits the representation G(D)= G(λ)dEλ,(2.12) and Dom(G(D)) =h∈ H:|G(λ)|2dEλ(h), h H<∞. The operator integrals (2.10) and (2.12) are understood as improper operator Stieltjes integrals which converge strongly (see, e.g., Section 8.2.1 in [34]). Let =(a,b],−∞ <a<b<∞,E:= Eb−Ea. The family Eof self-adjoint bounded non-negative operators from the Borel sets ⊆Rinto the space L( H)of bounded linear operators on a Hilbert space  His called an operator measure if E∪∞ j=1j=∞  j=1 Ej, where the limit at the right-hand side is understood in the sense of weakconvergence of operators, with i∩j=∅,i= j,E∅=0. From (iii), for every g,h∈ H,  dEλ(g), h H=g,h H  dEλ(h), h H=h2  H.(2.13) Thus, {Eλ,λ∈⊆R}provides a resolution of the identity. Note that, from (2.10) (see (i)–(v)), for every ψ∈Dom(D)⊆ H, D(ψ)2  H=D(ψ), D(ψ) H =DD(ψ), ψ H =|λ|2dEλ(ψ), ψ H =D λdEλ(ψ), ψ H= λdEλD(ψ), ψ H . (2.14) 123 M. D. Ruiz-Medina If D∈L( H), hence, Eq. (2.11) holds for every ψ∈ H,and from (2.14), D− λdEλ(ψ) 2  H =D(ψ) − λdEλ(ψ), D(ψ) − λdEλ(ψ) H =|λ|2dEλ(ψ), ψ H+|λ|2dEλ(ψ), ψ H−2|λ|2dEλ(ψ), ψ H=0. (2.15) Thus, D−λdEλL( H)=0,and the weak-sense representation (2.10) also holds in L( H)-norm. In particular, for D∈L0( H), with L0( H)denoting the class of compact operators on  H,the mapping λ−→ Eλhas discontinuities at the points given by the eigenvalues {λk(D), k≥1},with Eλk−lim  λ→λk(D); λ<λk(D) E λ=Pk, where Pkis the projection operator onto the eigenspace generated by the eigenvectors associated with the eigenvalue λk(D), for every k≥1. Let us now consider the following assumption: Assumption I. Assume that π −πFωL( H)dω<∞.(2.16) Remark 3 Assumption I holds, for instance, when the family {Fω,ω∈[−π,π]} is a.s. continuous in ω∈[−π, π],with respect to L( H)-norm, since applying reverse triangle inequality, FωL( H)is a.s. continuous in ω∈[−π,π]. Remark 4 Note that, under Assumption I, for every t∈Z, RtL( H)=π −π exp (iωt)FωdωL( H)≤π −πFωL( H)dω<∞.(2.17) The next preliminary result will be applied in the subsequent development. Lemma 1 Under Assumption I,  t∈ZRt2 S( H)=π −πFω2 S( H)dω<∞.(2.18) Proof Given an orthonormal basis {ψk,k≥1}of  H,under Assumption I,Fωis a.s. bounded in ω∈[−π,π].In particular, 123 Spectral analysis of multifractional LRD functional time series π −πFω(ψk), ψl Hdω≤ψl Hπ −πFω(ψk) Hdω≤π −πFωL( H)dω<∞, (2.19) for every k,l≥1.Hence, from (2.5) and (2.19), for every t∈Z, π −π exp (itω)Fω(ψk), ψl Hdω=Rt(ψk), ψl H,k,l≥1.(2.20) From (2.20), applying Fourier transform inversion formula,  t∈ZRt2 S(H)= t∈Z k,l≥1|Rt(ψk)(ψl)|2 = t∈Z k,l≥1π −ππ −π exp (it(ω −ξ))Fω(ψk)(ψl)Fξ(ψk)(ψl)dξdω = k,l≥1π −ππ −π t∈Z exp (it(ω−ξ))Fω(ψk)(ψl)Fξ(ψk)(ψl)dξdω = k,l≥1π −ππ −π δ(ω −ξ)Fω(ψk)(ψl)Fξ(ψk)(ψl)dξdω =π −π k,l≥1|Fω(ψk)(ψl)|2dω =π −πFω2 S( H)dω. (2.21) From Eqs. (2.20) and (2.21), keeping in mind that Fωis nonnegative symmetric operator,  t∈ZRt2 S(H)=π −πFω2 S( H)dω=π −πFωFωL1( H)dω ≤π −πFωL1( H)dω=π −π k≥1F ωFω1/2(ψk), ψk Hdω = k≥1π −πFω(ψk), ψk Hdω = k≥1R0(ψk), ψk H=R0L1( H)=σ2 X<∞.(2.22)  Remark 5 Under Assumption I, from Lemma 1,Fω∈S( H), for ω∈[−π,π]\0, with, as before, 0dω=0.Also, FωS( H)∈L2([−π,π],C). 123 M. D. Ruiz-Medina 3.4 Example 2. Discrete sampling of multifractional H-valued processes in continuous time Let H=L2(R,R), and  H=L2(R,C). Consider dEλ(ϕ), ψ H=R)ϕ(λ)) ψ(λ)dλ, (3.27) ) ψ(λ) =R exp (−iλ, z)ψ(z)dz,ψ∈L1(R), )ϕ(λ) =R exp (−iλ, z)ϕ(z)dz,ϕ∈L1(R). (3.28) With this particular choice, for (λ, ω) ∈R2,assume that the symbol f(ω,λ,θ) of the spectral density operator Fω,with respect to the spectral family {Eλ,λ∈R} introduced in (3.27)–(3.28) is defined as follows: f(ω,λ,θ)=|ω|−α(λ,θ)Nω(λ)h(ω), (3.29) where α(λ,θ) satisfies Assumption IV(i), and his a positive even taper function of bounded variation, with bounded support is the interval [−π, π],with h(−π) = h(π) =0 (see, e.g., [20]). We also assume that his Lipschitz-continuous function, and Nωis such that Mω,F(λ) =Nω(λ)h(ω) satisfies Assumption IV(ii). Furthermore, for ω∈[−π,π]\{0}, sup λ∈R|f(ω,λ,θ)|<∞.(3.30) As special case of (3.29), we can consider the tapered continuous version of Example 1 in Section 3.3 f(ω,λ,θ)=|ω|−α(λ,θ) P(λ, ω) Q(λ, ω)h(ω)1[−π,π](ω), (λ, ω) ∈R2, where the taper function satisfies the above required conditions, and Pand Qare positive polynomials such that Assumption IV(ii) holds. Particularly, when discrete sampling of the solution to fractional and multifractional pseudodifferential evolution equations with Gaussian functional innovations is considered, one can implement inference tools from this framework (see, e.g., [2,3,22]). 4 The convergence to zero in S(  H)norm of the bias of the integrated periodogram operator Theorem 1provides the convergence to zero, in the Hilbert–Schmidt operator norm, of the integrated bias of the periodogram operator. Note that, in [9], weak-convergence 123 Spectral analysis of multifractional LRD functional time series of the covariance operator of the fDFT to the spectral density operator, and the convergence of their respective traces is proved. The next result provides convergence in S( H)norm of the integrated covariance operator of the fDFT to the integrated spectral density operator, in the frequency domain, beyond the SRD condition assumed in [9]. Theorem 1 Under Assumption I, the following limit holds: π −πFω−F(T) ωdωS( H)→0,T→∞. Proof Let {ψk,k≥1}be an orthonormal basis of  H.Under Assumption I, π −πFω−F(T) ωdω 2 S( H) = k≥1π −ππ −πFξFω−FξF(T) ω−F(T) ξFω +F(T) ξF(T) ω(ψk)(ψk)dωdξ. (4.1) From Lemma 1(see Eq. (2.22)), for every k≥1,Fω(ψk)(ψk)∈L1([−π,π]). Hence, for each k≥1, F(T) ω(ψk)(ψk)→Fω(ψk)(ψk), T→∞,ω∈[−π, π]\0.(4.2) Applying triangle inequality, for every T≥2, ###Fω−F(T) ω(ψk)(ψk)###≤2FωL1( H)<∞, ω∈[−π,π]\0,(4.3) since from (2.22), π −πFωL1( H)dω<∞. Hence, from Eqs. (4.2)–(4.3), keeping in mind (2.22), Dominated Convergence Theorem leads to lim T→∞π −π###Fω−F(T) ω(ψk)(ψk)###dω =π −π lim T→∞###Fω−F(T) ω(ψk)(ψk)###dω=0,k≥1.(4.4) Note also that, from Lemma 1,FωS( H)∈L2([−π,π],C).Hence, for every k,l≥1,Fω(ψk)(ψl)∈L2([−π,π],C). Particularly, from Young’s convolution 123 M. D. Ruiz-Medina inequality with p=2,for each k,l≥1, π −π|F(T) ω(ψk)(ψl)|2dω≤π −π|Fω(ψk)(ψl)|2dω. (4.5) Hence, under Assumption I, applying the Cauchy–Schwarz and Jensen’s inequalities, and (4.5), from Lemma 1(see Eq. (2.22)), we obtain  k≥1π −ππ −π F(T) ξF(T) ω(ψk)(ψk)dωdξ = k≥1π −ππ −πF(T) ω(ψk), F(T) ξ(ψk) Hdωdξ ≤ k≥1π −πF(T) ω(ψk) Hdωπ −πF(T) ξ(ψk) Hdξ ≤4π2 k≥1*π −π F(T) ωF(T) ω(ψk)(ψk)dω*π −π F(T) ξF(T) ξ(ψk)(ψk)dξ ≤4π2+ , , - k≥1π −πF(T) ω(ψk)2  Hdω+ , , - k≥1π −πF(T) ξ(ψk)2  Hdξ =4π2+ , , - k,l≥1π −π|F(T) ω(ψk)(ψl)|2dω+ , , - k,l≥1π −π|F(T) ξ(ψk)(ψl)|2dξ ≤4π2+ , , - k,l≥1π −π|Fω(ψk)(ψl)|2dω+ , , - k,l≥1π −π|Fξ(ψk)(ψl)|2dξ =4π2π −πFω2 S( H)dω<∞.(4.6) Following similar steps to (4.6),  k≥1π −ππ −π F(T) ξFω(ψk)(ψk)dωdξ ≤4π2π −πFω2 S( H)dω<∞,(4.7) as well as  k≥1π −ππ −π FξF(T) ω(ψk)(ψk)dωdξ ≤4π2π −πFω2 S( H)dω<∞.(4.8) 123 Spectral analysis of multifractional LRD functional time series Furthermore, from Lemma 1(see Eq. (2.22)), Fω(ψk)(ψk)∈L1([−π,π]), for every k≥1.Thus, we can consider Young’s convolution inequality with p=1 leading to π −π F(T) ξ(ψk)(ψk)dξ≤π −π Fξ(ψk)(ψk)dξ, k≥1. Therefore, π −πF(T) ξL( H)dξ≤π −πFξL( H)dξ. (4.9) From Eqs. (4.6)–(4.8), we can apply Dominated Convergence Theorem, and keepinginmindEqs.(4.4) and (4.9), we obtain lim T→∞π −πFω−F(T) ωdω 2 S( H) = k≥1 lim T→∞π −ππ −πFξFω−FξF(T) ω−F(T) ξFω+F(T) ξF(T) ω(ψk)(ψk)dωdξ. ≤ k≥1 2π −πFξL( H)dξlim T→∞π −π###Fω−F(T) ω(ψk)(ψk)###dω=0.(4.10)  5 Semiparametric estimation in the spectral domain This section introduces the estimation methodology adopted in the functional spectral domain. Theorem 2derives the weak consistency of the formulated parametric estimator of the long-memory operator. Under Assumptions I-IV,let⊂Rp,p≥1,be a compact subset of Rp.Assume that the true parameter value θ0lies in the interior of , denoted as int . The symbol α:R×−→ (0,1)is such that α(·,θ 1)= α(·,θ 2), for θ1= θ2,for every θ1,θ 2∈. Thus, under (3.1), we get indentifiability in the semiparametric model. Denote by ) θT the estimator of the true parameter value θ0,based on a functional sample of size T. Hence, )αT(λ, θ) =α(λ,) θT)provides the parametric estimator of the symbol α(λ,θ) of Aθ. Let now introduce the elements involved in the definition of our operator loss function, to compute the minimum contrast estimator ) θT,under suitable conditions. Specifically, for each ω∈[−π,π],the weighting operator Wωis introduced as a bounded positive self-adjoint operator admitting the following spectral representation: 123 M. D. Ruiz-Medina Wω= W(ω,λ,β)dEλ,= W(λ)|ω|βdEλ,β>0.(5.1) In particular, the symbol W(ω,λ,β)of operator Wωfactorizes, in terms of  W(λ) and |ω|β,with  Wdefining the symbol of a positive self-adjoint operator  W∈L( H)such that m" W=inf ψ∈ H;ψ H=1 W(ψ), ψ H, M" W=sup ψ∈ H;ψ H=1 W(ψ), ψ H.(5.2) For each θ∈, the normalizing operator σ2 θis computed as follows: σ2 θ=π −π Fω,θWωdω=π −π Mω,F(λ)  W(λ) |ω|α(λ,θ)−βdEλdω. (5.3) Thus, the symbol 2 θof σ2 θis given by 2 θ(λ) =π −π Mω,F(λ)  W(λ) |ω|α(λ,θ)−βdω,∀λ∈. (5.4) Under Assumption IV(ii) (see Eq. (3.5)), and (5.2), for every λ∈, mm" W−1 −π+π 1|ω|−L(θ)+βdω+1 −1|ω|−l(θ)+βdω =mm" W(−π)1+β−L(θ) −(−1)1+β−L(θ) 1+β−L(θ) +(π)1+β−L(θ) −1 1+β−L(θ) +(−1)1−l(θ)+β 1−l(θ) +β +1 1−l(θ) +β ≤2 θ(λ) ≤MM" W−1 −π+π 1|ω|−l(θ)+βdω+1 −1|ω|−L(θ)+βdω =MM" W(−π)1+β−l(θ) −(−1)1+β−l(θ) 1+β−l(θ) +(π)1+β−l(θ) −1 1+β−l(θ) +(−1)1−L(θ)+β 1−L(θ) +β+1 1−L(θ) +β.(5.5) Thus, σ2 θis a bounded operator. The symbol of [σ2 θ]−1is given by 2 θ(λ)−1 =π −π Mω,F(λ)  W(λ) |ω|α(λ,θ)−βdω−1 ,λ∈. (5.6) 123 Spectral analysis of multifractional LRD functional time series From (5.5), for every λ∈, .mm" W(−π)1+β−L(θ) −(−1)1+β−L(θ) 1+β−L(θ) +(π)1+β−L(θ) −1 1+β−L(θ) +(−1)1−l(θ)+β 1−l(θ) +β+1 1−l(θ) +β/−1 ≥2 θ(λ)−1 ≥.MM" W(−π)1+β−l(θ) −(−1)1+β−l(θ) 1+β−l(θ) +(π)1+β−l(θ) −1 1+β−l(θ) +(−1)1−L(θ)+β 1−L(θ) +β+1 1−L(θ) +β/−1 .(5.7) Hence, [σ2 θ]−1is strictly positive and bounded. From (5.3), we can consider the following factorization of the spectral density operator, for (ω, θ) ∈[−π,π]\{0}×, Fω,θ =σ2 θϒω,θ =ϒω,θσ2 θ,(5.8) where, for each θ∈, and ω∈[−π,π],ω= 0, ϒω,θ = ϒ(ω,λ,θ)dEλ= Mω,F(λ) |ω|α(λ,θ)2 θ(λ)dEλ.(5.9) From Eqs. (5.1)–(5.9), for each θ∈, and any , ψ ∈ H, π −π ϒω,θWω()(ψ)dω= dEλ(), ψ H=, ψ H.(5.10) Equivalently, π −πϒω,θWωdωcoincides with the identity operator I Hon  H,for each θ∈. Let us now consider the empirical operator UT,θ given by, for each θ∈, [UT,θ ]=−π −π p(T) ωln ϒω,θWωdω, (5.11) where Tdenotes as before the sample size. Its theoretical counterpart Uθis defined, for each θ∈, as Uθ=−π −π Fω,θ0ln ϒω,θWωdω =−π −π Mω,F(λ)  W(λ) |ω|α(λ,θ0)−βln (ϒ(ω,λ,θ))dEλdω. (5.12) 123 M. D. Ruiz-Medina Remark 7 Note that, under Assumption IV(i), for each θ∈, Uθ∈L( H) for any β>0.Specifically, for every λ∈, Mω,F(λ) |ω|α(λ,θ0)∈L1([−π,π]), and ln (ϒ(ω,λ,θ))W(ω,λ,β)∈L1([−π,π]), with sup λ∈####−π −π Mω,F(λ) |ω|α(λ,θ0)ln (ϒ(ω,λ,θ))W(ω,λ,β)dω####<∞.(5.13) In addition, for Tlarge, UT,θ ∈L( H)a.s. (see Theorem 2below). We now consider the loss operator K(θ0,θ)to be minimized, with respect to θ, in the operator norm. Specifically, consider, for each θ∈, [K(θ0,θ)]=π −π Fω,θ0ln 0ϒω,θ0ϒ−1 ω,θ1Wωdω=[Uθ−Uθ0].(5.14) From Remark 7, for each θ∈, K(θ0,θ) ∈L( H). Furthermore, the symbol of K(θ0,θ)is given by π −π Mω,F(λ) |ω|α(λ,θ0)ln ϒ(ω,λ,θ0) ϒ(ω,λ,θ) W(ω,λ,β)dω, λ∈, θ ∈. (5.15) The operator [σ2 θ0]−1K(θ0,θ)could be interpreted as a weighted Kullback–Leibler divergence operator, measuring the discrepancy between the two semiparametric functional spectral models ϒω,θ0and ϒω,θ,for each θ∈(see, e.g., [11]). Note that, from Eqs. (5.3)–(5.14), applying Jensen’s inequality, for every k≥1,and θ∈, −[K(θ0,θ)](ψk)(ψk)≤σ2 θL( H) ln π −π ϒ(ω,λ,θ)W(ω,λ,β)dEλ(ψk), ψk Hdω =σ2 θL( H)ln π −π ϒω,θWω(ψk)(ψk)dω =σ2 θL( H)ln 0ψk2  H1=0.(5.16) for any orthonormal basis {ψk,k≥1}of  H.From Eq. (5.16), for every k≥1,and θ∈, [K(θ0,θ)](ψk)(ψk)≥0.Thus, {K(θ0,θ), θ ∈}is a parametric family of non-negative self-adjoint bounded operators such that K(θ0,θ)L( H)=sup k≥1[K(θ0,θ)](ψk)(ψk)>0,θ= θ0 K(θ0,θ)L( H)=sup k≥1[K(θ0,θ)](ψk)(ψk)=0⇔θ=θ0.(5.17) 123 Spectral analysis of multifractional LRD functional time series Hence, from Eqs. (5.17), θ0=arg min θ∈[K(θ0,θ)]L( H) =arg min θ∈sup k≥1 K(θ0, θ)(ψk)(ψk) =arg min θ∈sup k≥1 Uθ(ψk)(ψk). (5.18) We then consider the following estimator ) θTcomputed from the empirical contrast operator UT,θ in (5.11), and a given orthonormal basis {ψk,k≥1}of  H: ) θT=arg min θ∈sup k≥1 UT,θ (ψk)(ψk). (5.19) Theorem 2 Let {Xt,t∈Z}be a stationary zero-mean Gaussian functional sequence satisfying Assumptions I–IV. Consider in Assumption IV(ii) the particular case where Mω,Fsatisfies, for any ξ>0, lim ω→0Mω/ξ,FM−1 ω,F−I HL( H)=0.(5.20) Under the conditions reflected in Eqs. (5.1)–(5.16),forβ>1,we then have Eπ −πp(T) ω−Fω,θ0Wω,θdωS( H)→0,T→∞,(5.21) where, for (ω, θ) ∈[−π, π]×, Wω,θ =ln ϒω,θWω.(5.22) Furthermore, the estimator ) θTin (5.19) satisfies ) θT→Pθ0,T→∞, where →Pdenotes convergence in probability. Proof The operator Wω,θ introduced in (5.22) admits the spectral representation Wω,θ =ln (Mω(λ))−ln 02 θ(λ)1 −α(λ,θ) ln (|ω|)] W(λ)|ω|βdEλ,(5.23) 123 M. D. Ruiz-Medina for ω∈[−π,π],and θ∈. From (5.23), Wω,θL( H)≤ln Mω,F|ω|β WωL( H) +Aθln (|ω|)|ω|β WωL( H)+ln 0σ2 θ1|ω|β WωL( H) ≤ln(M)πβM" W+L(θ) ln(π)πβM" W+ln 0σ2 θ1L( H)πβM" W,(5.24) for every θ∈, ω ∈[−π,π],and β>0. From (5.24), sup ω∈[−π,π]Wω,θL( H)≤ln(M)πβM" W+L(θ) ln(π)πβM" W +ln 0σ2 θ1L( H)πβM" W=H(θ). (5.25) Thus, the family Wω,θ,ω∈[−π, π]!is equicontinuous, for any θ∈. We first prove that the following limits hold, for each θ∈, π −πF(T) ω,θ0−Fω,θ0Wω,θdωS( H)→0,T→∞ (5.26) Eπ −πp(T) ω−F(T) ω,θ0Wω,θdω 2 S( H)→0,T→∞,(5.27) where E0p(T) ω1=F(T) ω,θ0. From Theorem 1, and Eq. (5.26), π −πE0p(T) ω1−Fω,θ0Wω,θdωS( H) ≤H(θ) π −πF(T) ω,θ0−Fω,θ0dωS( H)→0,T→∞.(5.28) Under the Gaussian distribution of {Xt,t∈Z},applying Fourier Transform Inversion Formula, we obtain Eπ −πp(T) ω−F(T) ω,θ0Wω,θdω 2 S( H) = k≥1π −ππ −πEp(T) ξp(T) ω+F(T) ξ,θ0F(T) ω,θ0−F(T) ξ,θ0Ep(T) ω −Ep(T) ξF(T) ω,θ0W ξ,θWω,θ(ψk)(ψk)dωdξ = k≥1π −ππ −πEp(T) ξp(T) ω−F(T) ξ,θ0F(T) ω,θ0W ξ,θWω,θ(ψk)(ψk)dωdξ 123 Spectral analysis of multifractional LRD functional time series =1 (2πT)2 k≥1π −ππ −π⎡ ⎣ T  t1,s1,t2,s2=1 exp (−iω(t1−s1)−iξ(t2−s2)) ×EXt1⊗Xs1⊗Xt2⊗Xs2−EXt1⊗Xs1EXt2⊗Xs2 ×W ξ,θWω,θ(ψk)(ψk)dωdξ =1 (2πT)2 k≥1π −ππ −π⎡ ⎣ T  t1,s1,t2,s2=1 exp (−iω(t1−s1)−iξ(t2−s2)) ×EXt1⊗Xt2EXs1⊗Xs2+EXt1⊗Xs2EXt2⊗Xs1 ×W ξ,θWω,θ(ψk)(ψk)dωdξ =2π T k≥1π −ππ −ππ −ππ −π Fω,θ0F ξ,θ0⎡ ⎣1 [2π]3T T  t1,s1,t2,s2=1 exp (it1(ω−ω)) ×exp is1(ω + ξ)+it2(−ξ−ω) +is2(ξ − ξ) +exp it1(ω−ω) +is1(ω + ξ)+it2(−ξ− ξ)+is2(ξ −ω) ×W ξ,θWω,θ(ψk)(ψk)dωdξdωd ξ =2π T k≥1π −πf2(ω) f1(ω) g2(ω) g1(ω) h2(ω,u1) h1(ω,u1) 4 T(u1,u2,u3)Fu1+ω,θ0Fu2−ω,θ0 ×W −(u1+u3+ω),θ Wω,θ(ψk)(ψk)du3du2du1dω +2π T k≥1π −πf2(ω) f1(ω) g2(ω) g1(ω)  h2(ω,u1)  h1(ω,u1) 4 T(u1,u2,u3)Fu1+ω,θ0Fu2−ω,θ0 ×W u3−u1−ω,θWω,θ(ψk)(ψk)du3du2du1dω =2π Tπ −πf2(ω) f1(ω) g2(ω) g1(ω) h2(ω,u1) h1(ω,u1) 4 T(u1,u2,u3) ×Fu1+ω,θ0Wω,θ,Fu2−ω,θ0W−(u1+u3+ω),θ S( H)du3du2du1dω +2π Tπ −πf2(ω) f1(ω) g2(ω) g1(ω)  h2(ω,u1)  h1(ω,u1) 4 T(u1,u2,u3) ×Fu1+ω,θ0Wω,θ,Fu2−ω,θ0Wu3−u1−ω,θS( H)du3du2du1dω ≤Kπ T[−π,π]4 4 4T(u1,u2,u3) ×F2u1+ω,θ0Wω,θ,F2u2−ω,θ0W−(2u1+4u3+ω),θ S( H)dωdu1du2du3 +Kπ T[−π,π]4 4 4T(u1,u2,u3) ×F2u1+ω,θ0Wω,θ,F2u2−ω,θ0W4u3−2u1−ω,θS( H)dωdu1du2du3,(5.29) where, for ω∈[−π,π],f1(ω) =−π−ω, f2(ω) =π−ω, g1(ω) =−π+ω, g2(ω) =π+ω, h1(ω, u1)=−π−u1−ω, h2(ω, u1)=π−u1−ω, h1(ω,u1)= 123 M. 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