VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA170 10.15559/20-VMSTA170 Research Article Asymptotic normality of the residual correlogram in the continuous-time nonlinear regression model https://orcid.org/0000-0001-5250-6781 IvanovAlexanderalexntuu@gmail.com∗ https://orcid.org/0000-0002-1870-9873 MoskvychovaKaterynamoskvychova@matan.kpi.ua National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Peremohy avenue 37, Kyiv, Ukraine Corresponding author. 2021 211220208193113 2872020 5122020 5122020 © 2021 The Author(s). Published by VTeX2021 Open access article under the CC BY license.

In a continuous time nonlinear regression model the residual correlogram is considered as an estimator of the stationary Gaussian random noise covariance function. For this estimator the functional central limit theorem is proved in the space of continuous functions. The result obtained shows that the limiting sample continuous Gaussian random process coincides with the limiting process in the central limit theorem for standard correlogram of the random noise in the specified regression model.

 Nonlinear regression model stationary Gaussian noise covariance function residual correlogram asymptotic normality 62J02 62F12 62M10 Introduction

Estimation of the signal parameters in the “signal+noise” observation model is a classic problem of statistics of stochastic processes. If the signal (regression function) nonlinearly depends on parameters, then this is a problem of nonlinear time-series regression analysis. Another problems arise when there is a need to estimate the functional characteristics of the correlated random noise in the given functional regression model. For the stationary noise it can be estimation of the noise spectral density or covariance function. Asymptotic properties of the Whittle and Ibragimov estimators of spectral density parameters in the continuous time nonlinear regression model were considered in Ivanov and Prykhod’ko [16, 15], Ivanov et al. [17]. Exponential bounds for the probabilities of large deviations of the stationary Gaussian noise covariance function in the similar regression model are obtained in Ivanov et al. [11]. Stochastic asymptotic expansion and asymptotic expansions of the bias and variance of the residual correlogram in the same setting were derived in Ivanov and Moskvychova [21, 19]. In both cases it is first necessary to estimate the parameters of the regression function to neutralize it influence, and then use residual periodogram to estimate spectrum parameters and residual correlogram to estimate covariance function. The residual correlogram generalizes the notion of the averaged residual sum of squares in classical regression analysis.

However, unlike the residual sum of squares and usual correlogram, the results on the residual correlogram are not sufficiently represented in statistical literature except for a few theorems for discrete time linear regression with stationary correlated observation errors (see Anderson [1], Hannan [8]). These statements were obtained using explicit expressions for the least squares estimator (LSE) of unknown regression parameters. In the multitude of works dealing with stationary stochastic processes in the correlograms the values of the processes are centered by their sample means that are the LSE of their expectations. Some field generalizations of such a centering can be found in Leonenko [22].

In this paper we prove the functional central limit theorem (CLT) in the space of continuous functions for the normed residual correlogram as an estimator of the stationary Gaussian random noise covariance function in continuous time nonlinear regression model. The first result of such a kind has been obtained in Ivanov and Moskvychova [20]. In current paper we significantly weakened the requirements to the regression function under which the indicated CLT is true, namely: brought them closer to the conditions of the LSE asymptotic normality [18]. In addition we replaced the condition for the existence of a certain moment of the noise spectral density by much weaker condition on the weighted spectral density admissibility with respect to regression function spectral measure. In the last section of the paper we apply our result to the trigonometric regression.

 Setting

Suppose the observations are of the form X(t)=g(t,θ0)+ε(t),t∈[0,+∞), where g:(−γ,+∞)×Θγ→R is a continuous function depending on unknown parameter θ0=(θ10,…,θq0)∈Θ⊂Rq , Θ is an open convex set, Θγ=⋃‖e‖≤1(Θ+γe) , γ is some positive number, and ε is a random noise described below.

The assumption about domain (−γ,+∞) for function g in t is of technical nature and does not affect possible applications. This assumption makes it possible to formulate the condition RN1(i) which is used in the proof of Lemma 3.

N1. (i) ε=ε(t),t∈R is a real sample continuous stationary Gaussian process defined on a complete probability space Ω,F,P , Eε(0)=0 ;

(ii) covariance function B=B(t),t∈R of the process ε is absolutely integrable.

Obviously, if B∈L1(R) , then the process ε has a bounded and continuous spectral density f=f(λ),λ∈R .

LSE of unknown parameter θ0∈Θ obtained by observations of the process X(t),t∈[0,T] is said to be any random vector θˆT=(θˆ1T,…,θˆqT)∈Θc ( Θc is the closure of Θ in Rq ) such that QT(θˆT)=minτ∈ΘcQT(τ),QT(τ)= ∫0TX(t)−g(t,τ)2dt, provided that the minimum in (2) is attained a.s.

The existence of at least one such a vector follows from the Pfanzagl results [23].

As an estimator of B we take the residual correlogram built by residuals Xˆ(t)=X(t)−g(t,θˆT),t∈[0,T+H], namely: BT(z,θˆT)=T−1 ∫0TXˆ(t+z)Xˆ(t)dt,z∈[0,H], H>0 0$]]> is some fixed number. In particular BT(0,θˆT)=T−1QT(θˆT) is LSE of the variance B(0) of stochastic process ε. On the other hand BT(z,θ0)=BT(z)=T−1 ∫0Tε(t+z)ε(t)dt,z∈[0,H], is the correlogram of the process ε.

From the condition N1 it follows that integrals (3) and (4) can be considered as Riemann integrals based on single paths of the corresponding processes and BT(z,θˆT) , BT(z) , z∈[0,H] , are sample continuous stochastic processes.

Consider the normalized residual correlogram XT(z)=T1/2BT(z,θˆT)−B(z)=YT(z)+RT(z),z∈[0,H],YT(z)=T1/2BT(z)−B(z),z∈[0,H],RT(z)=T−1/2I1T(z)+T−1/2I2T(z)+T−1/2I3T(z),z∈[0,H], with I1T(z)= ∫0T(g(t+z,θˆT)−g(t+z,θ))(g(t,θˆT)−g(t,θ))dt, I2T(z)= ∫0Tε(t+z)(g(t,θˆT)−g(t,θ))dt, I3T(z)= ∫0Tε(t)(g(t+z,θˆT)−g(t+z,θ))dt.

We will consider the processes XT , YT , and RT as random elements in the measurable space C([0,H]),B of continuous functions on [0,H] with Borel σ-algebra B .

Let Z be a random element in the indicated space. The distribution of Z is the probability measure PZ−1 on C([0,H]),B .

A family UT of random elements converges in distribution, as T→∞ , to a random element U in the space C([0,H]) (we write UT⟶DU ), if the distributions PUT−1 of elements UT converge weakly, as T→∞ , to the distribution PU−1 of the element U.

Since f∈L2(R) under assumption N1(ii), as it is well known, for any z1,z2∈[0,H] , as T→∞ , EYT(z1)YT(z2)⟶b(z1,z2)=4π∫Rf2(λ)cosλz1cosλz2dλ. and (see, e.g., Buldygin [3]) all the finite-dimensional distributions of the processes YT weakly converge, as T→∞ , to the Gaussian process Y with zero mean and covariance function (9).

We assume that the process Y is separable.

Introduce the function (see section 6.4 of the chapter 6 in Buldygin and Kozachenko [4]) q(z)=∫Rf2(λ)sin2λz2dλ1/2,h≥0.

If f∈L2(R) , the function q generates pseudometrics ρ(z1,z2)=q(|z1−z2|),ρ(z1,z2)=ρ(z1,z2),z1,z2∈R. Denote by Hρ(ε)=Hρ([0,1],ε) , ε>0 0$]]>, the metric entropy of the interval [0,1] generated by the pseudometric ρ , ∫0+ the integral over an arbitrary neighborhood of zero (0,δ) , δ>0 0$]]>.

Below we are going to formulate a theorem obtained in Buldygin and Kozachenko [4] (Theorem 6.4.1) under milder conditions than ours. In the absence of assumption on sample continuity of the process ε from the condition f∈L2(R) it follows that correlograms can be understood, as continuous in probability with respect to the parameter z Riemann meansquare integrals. Due to Lemma 6.4.1 in [4] we can conclude that processes YT,T>0 0$]]>, are likewise continuous in probability. Thus, it can be assumed that the processes YT,T>0 0$]]>, are separable.

Let f∈L2(R) and N2.∫0+Hρ(ε)dε<∞. Then for any H>0 0$]]> I) Y∈C([0,H]) a.s.; II) YT∈C([0,H]) a.s., T>0 0$]]>; III)   YT⟶DY , as T→∞ , in the space C([0,H]) .

In particular, for any x>0 0$]]> limT→∞Psupz∈[0,H]|YT(z)|>x=Psupz∈[0,H]|Y(z)|>x. x\right\}=\mathsf{P}\left\{\underset{z\in [0,H]}{\sup }|Y(z)|>x\right\}.\]]]>

The conclusion of the Theorem 1 is true under conditions N1 and N2. (see Theorem 6.4.1 in [11]).

As it is shown in the Remark 6.4.1 in [4] the condition N2 is satisfied if for some δ>0 0$]]> ∫0∞f2(λ)ln4+δ(1+λ)dλ<∞. In turn (10) follows from the condition f∈L2(R) under natural restrictions on the decreasing of the spectral density f at infinity (see Theorem 6.4.2 in [4])

Taking into account the Theorem 1, we state a simple but important fact that is a rephrasing for C([0,H]) of the Theorem 3.1 in Billingsley [2], p. 27. For functions a(z),z∈[0,H] , we will write ‖a‖=supz∈[0,H]|a(z)| .

If YT⟶DY and RT⟶P0,asT→∞, then XT⟶DY , as T→∞ .

Thus, to obtain a functional theorem in C([0,H]) on asymptotic normality of the normalized residual correlogram XT it is required to prove (11).

 Conditions

To prove (11) we need some regularity conditions imposed on the regression function g, spectral density f and LSE θˆT .

Assume that for any t>−γ -\gamma $]]> the function g(t,θ) is twice continuously differentiable with respect to θ∈Θγ , and moreover, the derivatives gi(t,θ)=∂/∂θig(t,θ) , gij(t,θ)=∂2/∂θi∂θjg(t,θ) , i,j=1,q‾ , are continuous in the totality of variables. Denote dT(θ)=diagdiT(θ),i=1,q‾,diT2(θ)= ∫0Tgi2(t,θ)dt,θ∈Θ,i=1,q‾, and suppose that lim infT→∞T−1diT2(θ)>0,θ∈Θ,i=1,q‾, 0,\hspace{2.5pt}\theta \in \Theta ,\hspace{2.5pt}i=\overline{1,q},\]]]> in particular, these limits can be infinite. Let also dij,T(θ)= ∫0Tgij2(t,θ)dt,θ∈Θ,i,j=1,q‾.

Introduce now the normalized LSE uˆT=dT(θ0)θˆT−θ0 , θ0∈Θ , and the notation h(t,u)=g(t,θ0+dT−1(θ0)u) , hi(t,u)=gi(t,θ0+dT−1(θ0)u) , hij(t,u)=gij(t,θ0+dT−1(θ0)u) , u∈UT(θ0)=dT(θ0)Θc−θ0 , i,j=1,q‾ ; v(r)=u∈Rq:‖u‖<r ; ΦTθ1,θ2= ∫0Tg(t,θ1)−g(t,θ2)2dt,θ1,θ2∈Θc;ΨiTz1,z2;θ= ∫0Tgi(t+z1,θ)−gi(t+z2,θ)2dt,z1,z2≥0,θ∈Θc,i=1,q‾.

Instead of the words “for all sufficiently large T” we will write below “for T>T0 {T_{0}}$]]>”. Assume that the following conditions are satisfied.

R1. There exists a constant k0<∞ such that for any θ0∈Θ and T>T0 {T_{0}}$]]>, where k0 and T0 may depend on θ0 , ΨT(θ,θ0)≤k0‖dT(θ0)(θ−θ0)‖2,θ∈Θc.

R2. For any r≥0 θ0∈Θ , and T>T0(r) {T_{0}}(r)$]]>

(i) diT−1(θ0)supt∈[0,T],u∈Vc(r)∩UT(θ0)|hi(t,u)|≤ki(r)T−1/2 , i=1,q‾ ;

(ii) dij,T−1(θ0)supt∈[0,T],u∈Vc(r)∩UT(θ0)|hij(t,u)|≤kij(r)T−1/2 , i,j=1,q‾ ;

(iii) diT−1(θ0)djT−1(θ0)dij,T−1(θ0)≤k˜ijT−1/2 , i,j=1,q‾ , with constants ki , kij , k˜ij , possibly, depending on θ0 .

R3. There exist constants ki<∞ , i=1,q‾ , such that for any θ0∈Θ and T>T0 {T_{0}}$]]>, where k‾i and T0 may depend on θ0 , diT−2(θ0)ΨiT(z1,z2;θ0)≤k‾i|z1−z2|2,z1,z2∈[0,H].

If condition R2(i) is fulfilled for r=0 , then for any fixed H>0 0$]]> and θ0∈Θ di,T+H(θ0)diT−1(θ0)⟶1,asT→∞,i=1,q‾.

We have qi=di,T+H2(θ0)diT−2(θ0)=1+ ∫TT+Hgi2(t,θ0)dtdiT−2(θ0)≤≤1+Hsup0≤t≤T+H|gi(t,θ0)|2di,T+H−2(θ0)qi≤1+ki2(0)HT+Hqi. Then for T>T0 {T_{0}}$]]> 1≤qi≤11−ki2(0)H(T+H)−1,andqi→1,asT→∞,i=1,q‾.  □

For basic observation model (1), we introduce a family of matrix-valued measures μT(dλ;θ) , θ∈Θ , T>0 0$]]>, on R,L(R) , L(R) is the σ-algebra of Lebesgue measurable subsets of R , with matrix densities with respect to Lebesgue measure (μTjl(λ,θ))j,l=1q , θ∈Θ , μTjl(λ,θ)=gTj(λ,θ)gTl(λ,θ)‾∫RgTj(λ,θ)2dλ∫RgTl(λ,θ)2dλ−1/2,gTj(λ,θ)= ∫0Teiλtgj(t,θ)dt,j,l=1,q‾. by Plancherel identity ∫RgTj(λ,θ)gTl(λ,θ)‾dλ=2π ∫0Tgj(t,θ)gl(t,θ)dt, in particular, djT2(θ)=(2π)−1∫RgTj(λ,θ)2dλ,j,l=1,q‾.

R4. The family of measures μT(dλ;θ) converges weakly to a positive definite matrix measure μ(dλ;θ)=μjl(dλ;θ)j,l=1q , as T→∞ , θ∈Θ .

Condition R4 means that the elements μjl(dλ;θ) of the matrix measure μ(dλ;θ) are complex signed measures of bounded variation and the matrix μ(A;θ) is positive semi-definite for any A∈L , and μ(R;θ) is a positive definite matrix, θ∈Θ .

The measure μ(dλ;θ),θ∈Θ , is called the spectral measure of the regression function g(t,θ) , or, more precisely, the spectral measure of the gradient ∇g(t,θ),θ∈Θ , see Grenander [6], Holevo [9], Ibragimov and Rozanov [10], and Ivanov and Leonenko [14].

Taking into account (16), (17) and condition R4 we get μT(R;θ)=∫RμT(dλ;θ)=djT−1(θ)dlT−1(θ) ∫0Tgj(t,θ)gl(t,θ)dtj,l=1q==JT(θ)⟶J(θ)=∫Rμ(dλ;θ)>0,asT→∞,θ∈Θ. 0,\hspace{2.5pt}\mathrm{as}\hspace{2.5pt}T\to \infty ,\hspace{2.5pt}\theta \in \Theta .\end{array}\]]]>

AN. The random vector dT(θ0)(θˆT−θ0) is asymptotically, as T→∞ , normal with zero mean and covariance matrix ΣLSE=2π∫Rμ(dλ;θ0)−1∫Rf(λ)μ(dλ;θ0)∫Rμ(dλ;θ0)−1.

Sufficient conditions of the assumption AN fulfillment are bulky. These conditions are given in [14] and, for example, in Ivanov et al. [18]. At least, conditions R2 and R4 form the part of these conditions in [18].

Consider the diagonal elements (measures) μjj,j=1,q‾ , of the matrix spectral measure μ.

(Billingsley [<xref ref-type="bibr" rid="j_vmsta170_ref_002">2</xref>], Ibragimov and Rozanov [<xref ref-type="bibr" rid="j_vmsta170_ref_010">10</xref>]).

A function b(λ) , λ∈R , is called μjj -admissible, if it is integrable with respect to μjj and ∫Rb(λ)μTjj(dλ;θ)⟶∫Rb(λ)μjj(dλ;θ),asT→∞,θ∈Θ.

RN. For some δ∈(0,1] the function b(λ)=|λ|1+δf(λ) , λ∈R , is μjj -admissible, j=1,q‾ .

Consider some sufficient conditions on μjj -admissibility of the function b from assumption RN under condition N1(ii). N3.supλ∈Rλ1+δf(λ)<∞. Under condition N3 the relation (18) follows from N1(ii) and definition of weak convergence. Denote (∂/∂t)gj(t,θ)=gj′(t,θ),g˜Tj(λ,θ)= ∫0Teiλtgj′(t,θ)dt and introduce the next condition

RN1. (i) The functions gj(t,θ),θ∈Θ , are continuously differentiable with respect to t>−γ -\gamma $]]>, and there exists λ0=λ0(θ)>0 0$]]> such that for T>T0(θ) {T_{0}}(\theta )$]]> sup|λ|>λ0djT−2(θ)|g˜Tj(λ,θ)|2≤hj(θ)<∞,j=1,q‾,θ∈Θ. {\lambda _{0}}}{\sup }{d_{jT}^{-2}}(\theta )|{\tilde{g}_{T}^{j}}(\lambda ,\theta ){|^{2}}\le {h_{j}}(\theta )<\infty ,\hspace{2.5pt}j=\overline{1,q},\hspace{2.5pt}\theta \in \Theta .\]]]>

(ii) There exists λ1>0 0$]]> such that for λ>λ1 {\lambda _{1}}$]]> the function λ1+δf(λ) strictly increases, and |λ|1+δf(λ)⟶∞,asλ→∞. (iii)∫R|λ|1+δf(λ)μjj(dλ;θ)<∞,j=1,q‾,θ∈Θ.

Conditions N1(ii), RN1, R2(i) fulfilled for r=0 , and R4 imply the conditions RN.

For M>0 0$]]> consider the cut-off function bM(λ)=b(λ)χ{λ:b(λ)≤M}+Mχ{λ:b(λ)>M},∫Rb(λ)μTjj(dλ;θ)−∫Rb(λ)μjj(dλ;θ)≤≤∫Rb(λ)μTjj(dλ;θ)−∫RbM(λ)μTjj(dλ;θ)++∫RbM(λ)μTjj(dλ;θ)−∫RbM(λ)μjj(dλ;θ)++∫RbM(λ)μjj(dλ;θ)−∫Rb(λ)μjj(dλ;θ)==K1j(T,M)+K2j(T,M)+K3j(M),j=1,q‾. M\},\\ {} \displaystyle \left|\underset{\mathbb{R}}{\int }b(\lambda ){\mu _{T}^{jj}}(d\lambda ;\theta )-\underset{\mathbb{R}}{\int }b(\lambda ){\mu ^{jj}}(d\lambda ;\theta )\right|\le \\ {} \displaystyle \le \left|\underset{\mathbb{R}}{\int }b(\lambda ){\mu _{T}^{jj}}(d\lambda ;\theta )-\underset{\mathbb{R}}{\int }{b^{M}}(\lambda ){\mu _{T}^{jj}}(d\lambda ;\theta )\right|+\\ {} \displaystyle +\left|\underset{\mathbb{R}}{\int }{b^{M}}(\lambda ){\mu _{T}^{jj}}(d\lambda ;\theta )-\underset{\mathbb{R}}{\int }{b^{M}}(\lambda ){\mu ^{jj}}(d\lambda ;\theta )\right|+\\ {} \displaystyle +\left|\underset{\mathbb{R}}{\int }{b^{M}}(\lambda ){\mu ^{jj}}(d\lambda ;\theta )-\underset{\mathbb{R}}{\int }b(\lambda ){\mu ^{jj}}(d\lambda ;\theta )\right|=\\ {} \displaystyle ={K_{1j}}(T,M)+{K_{2j}}(T,M)+{K_{3j}}(M),\hspace{2.5pt}j=\overline{1,q}.\end{array}\]]]>

By Lebesgue monotonic convergence theorem from RN1(iii) we get K3j(M)⟶0,asM→∞.

Under conditions N1(ii) and R4 for any fixed M>0 0$]]> K2j(T,M)⟶0,asT→∞.

On the other hand, K1j(T,M)=(2π)−1∫{λ:b(λ)>M}(b(λ)−M)djT−2(θ)|gTj(λ,θ)|2dλ≤≤(2π)−1∫{λ:b(λ)>M}b(λ)djT−2(θ)|gTj(λ,θ)|2dλ. M\}}{\int }(b(\lambda )-M){d_{jT}^{-2}}(\theta )|{g_{T}^{j}}(\lambda ,\theta ){|^{2}}d\lambda \le \\ {} \displaystyle \le {(2\pi )^{-1}}\underset{\{\lambda :b(\lambda )>M\}}{\int }b(\lambda ){d_{jT}^{-2}}(\theta )|{g_{T}^{j}}(\lambda ,\theta ){|^{2}}d\lambda .\end{array}\]]]> Integrating by parts we obtain (see (15) and RN1(i)) |gTj(λ,θ)|=|λ|−1eiλTgj(T,θ)−gj(0,θ)−g˜Tj(λ,θ)≤≤|λ|−12supt∈[0,T]|gj(t,θ)|+|g˜Tj(λ,θ)|. Thus under condition R2(i) with r=0 djT−2(θ)gTj(λ,θ)2≤λ−24djT−2(θ)supt∈[0,T]gj(λ,θ)2+2dT−2(θ)g˜Tj(λ,θ)2K1j(T,M)≤2kj2(0)π−1T−1∫{λ:b(λ)>M}|λ|−1+δf(λ)dλ++π−1∫{λ:b(λ)>M}|λ|−1+δf(λ)djT−2(θ)g˜Tj(λ,θ)2dλ==K1j(1)(T,M)+K1j(2)(T,M). M\}}{\int }|\lambda {|^{-1+\delta }}f(\lambda )d\lambda +\\ {} \displaystyle +{\pi ^{-1}}\underset{\{\lambda :b(\lambda )>M\}}{\int }|\lambda {|^{-1+\delta }}f(\lambda ){d_{jT}^{-2}}(\theta ){\left|{\widetilde{g}_{T}^{j}}(\lambda ,\theta )\right|^{2}}d\lambda =\\ {} \displaystyle ={K_{1j}^{(1)}}(T,M)+{K_{1j}^{(2)}}(T,M).\end{array}\]]]> Let ε>0 0$]]> be an arbitrary fixed number. Since integral in K1j(1)(T,M) is majorized by the spectral moment ∫R|λ|−1+δf(λ)dλ<∞ , then for T>T0 {T_{0}}$]]> we have K1j(1)(T,M)≤ε/4 .

Put maxλ∈[0,λ1]b(λ)=b(λ¯1) for some λ¯1≥λ1 , where λ1 is the number from the condition RN1(ii). Let also Λ>0 0$]]> is such a number that b(λ)>M=b(Λ)>b(λ¯1) M=b(\Lambda )>b({\bar{\lambda }_{1}})$]]>. In this case {λ:b(λ)>M}={λ:|λ|>Λ} M\}=\{\lambda :|\lambda |>\Lambda \}$]]>, and if Λ≥λ0 , then for T>T0 {T_{0}}$]]> from the condition RN1(i) we get K1j(2)(T,M)=π−1∫{λ:|λ|>Λ}|λ|−1+δf(λ)djT−2(θ)|g˜Tj(λ,θ)|2dλ≤≤π−1hj(θ)∫{λ:|λ|>Λ}|λ|−1+δf(λ)dλ. \Lambda \}}{\int }|\lambda {|^{-1+\delta }}f(\lambda ){d_{jT}^{-2}}(\theta )|{\tilde{g}_{T}^{j}}(\lambda ,\theta ){|^{2}}d\lambda \le \\ {} \displaystyle \le {\pi ^{-1}}{h_{j}}(\theta )\underset{\{\lambda :|\lambda |>\Lambda \}}{\int }|\lambda {|^{-1+\delta }}f(\lambda )d\lambda .\end{array}\]]]> Now by the choice of Λ, that is by the choice of cut-off level M=b(Λ) , we get the inequality K1j(2)(T,M)<ε/4 . Increasing Λ if necessary, from (21) we obtain K3j(T,M)<ε/4 . As well increasing T0 if necessary, we receive from (22) K2j(T,M)<ε/4 .  □

 Asymptotic normality of the residual correlogram

In this section, we formulate and prove the CLT for the normalized residual correlogram XT(z),z∈[0,H] in the Banach space of continuous functions C[0,H] with uniform norm.

If the conditions N1, N2, R1–R4, AN, and RN are satisfied, then XT(·)=T1/2BT·,θˆT−B(·)⟶DY,asT→∞.

In view of the Theorem 1 and Lemma 1 of Section 2, to obtain (23) it is sufficient to prove (11). So, taking into account the expressions (5)–(8), the proof of the Theorem 2 consists of 3 lemmas.

If conditions R1, R2(i) for r=0 , and AN are fulfilled, then T−1/2I1T⟶P0,asT→∞.

Obviously, by conditions R1, Lemma 2 T−1/2I1T≤T−1/2ΦT1/2(θˆT,θ0)ΦT+H1/2(θˆT,θ0)≤≤k0max1≤j≤qdi,T+H(θ0)di,T−1(θ0)T−1/2dT(θ0)(θˆT−θ0)·dT(θ0)(θˆT−θ0), and T−1/2I1T⟶P0,asT→∞ , due to the condition AN.  □

We will use the notation αiT=diT(θ0)(θˆiT−θi0) , i=1,q‾ .

Under conditions N1, R2(ii), R2(iii), R4, RN, and AN T−1/2I2T⟶P0,asT→∞.

Apply the Taylor formula to the integral T−1/2I2T and write T−1/2I2T= ∑j=1qdjT−1(θ0) ∫0Tε(t+z)gj(t,θ0)dtT−1/2αjT++12 ∑i,j=1qdiT−1(θ0)djT−1(θ0) ∫0Tε(t+z)gij(t,θT∗)dtαiTT−1/2αjT==∑1,T(z)+12∑2,T(z),θT∗=θ+ηθˆT−θ0,η∈(0,1)a.s. Consider sample continuous Gaussian stochastic processes ξjT(z)=djT−1(θ0) ∫0Tε(t+z)gj(t,θ0)dt,z∈[0,H],T>0,j=1,q‾. 0,\hspace{2.5pt}\hspace{2.5pt}j=\overline{1,q}.\]]]>

Subject R4, as T→∞ , BjT(z1−z2)=EξjT(z1)ξjT(z2)==djT−2(θ0) ∫0T ∫0TB(t−s+z1−z2)gj(t,θ0)gj(s,θ0)dtds==2π∫Reiλ(z1−z2)f(λ)μTjj(dλ,θ0)⟶2π∫Rcosλ(z1−z2)f(λ)μjj(dλ,θ0)==Bj(z1−z2),z1,z2∈[0,H].

Thus all finite-dimensional distributions of the stationary Gaussian processes ξjT(z),z∈[0,H] converge to the corresponding finite-dimensional distributions of the stationary Gaussian processes ξj=ξj(z),z∈[0,H] with covariance functions Bj(z) , z∈[0,H] , j=1,q‾ . We assume, that the processes ξj,j=1,q‾ , are separable.

Since by condition RN for some δ∈(0,1] kj(δ,θ0)=∫R|λ|1+δf(λ)μjj(λ,θ0)<∞,j=1,q‾, then Eξj(z1)−ξj(z2)2=2Bj(0)−Bj(z1−z2)≤≤22−δπkj(δ,θ0)|z1−z2|1+δ,z1,z2∈[0,H].

According to the Kolmogorov theorem (see, for example, Gikhman and Skorokhod [5]) the processes ξj are sample continuous. Moreover, under condition RN for T>T0 {T_{0}}$]]> EξjT(z1)−ξjT(z2)2=2BjT(0)−BjT(z1−z2)≤≤22−δπkj(δ,θ0)+1|z1−z2|1−δ. So, ξjT⟶Dξj,asT→∞j=1,q‾ , in the space C([0,H]) and (see again [5]) for all continuous on C([0,H]) functionals ℓ the distribution of ℓ(ξjT) converges, as T→∞ , to the distribution of ℓ(ξj) . Using the same notation for weak convergence of random variables, in particular, we obtain ξjT⟶Dξj,j=1,q‾ , and (see (24)) ‖∑1T‖≤ ∑j=1qξjTT−1/2|αjT|⟶P0,asT→∞.

Let sT∗=T−1∫0Tε2(t)dt . Then ∫02T|ε(t)|dt≤T(1+s2T∗).

Note that ‖dT(θ)(θT∗−θ0)‖≤‖dT(θ0)(θˆT−θ0)‖ , and if the events AT(r)=dT(θ0)(θˆT−θ0)≤r,AT∗=s2T∗≤1+B(0) occur, then supt∈[0,T]gij(t,θT∗)≤supt∈[0,T],u∈Vc(r)∩UT(θ0)hij(t,u), and by the conditions R2(ii), R2(iii) for the norm of any term ∑2,Tij of the sum ∑2,T we get the upper bound ∑2,Tij≤T1/2diT−1(θ0)djT−1(θ0)dij,T(θ0)(1+s2T∗)××T1/2dij,T−1(θ0)supt∈[0,T],u∈Vc(r)∩UT(θ0)hij(t,u)|αiT|(T−1/2|αjT|)≤≤r2kij(r)k˜ij(r)(2+B(0))T−1/2. Under condition AN for any δ>0 0$]]> it is possible to find r>0 0$]]> such that for T>T0(δ) {T_{0}}(\delta )$]]> PAT(r)‾<δ. On the other hand, by Isserlis’ theorem (see, for example, [14]) PAT∗‾≤Es2T∗−B(0)2=2(2T)−2 ∫02T ∫02TB2(t−s)dtds≤≤‖B‖22T−1,‖B‖2=∫RB2(t)dt1/2<∞.

From the inequalities (27)–(29) it follows ∑2,T⟶P0,asT→∞.  □

Under conditions N1, R2–R4, RN, and AN T−1/2I3T⟶P0,T→∞.

We write T−1/2I3T(z)= ∑j=1qdjT−1(θ0) ∫0Tε(t)gj(t+z,θ0)dtT−1/2αjT++12 ∑i,j=1qdiT−1(θ0)djT−1(θ0) ∫0Tε(t)gij(t+z,θT∗)dtαiTT−1/2αjT==∑3,T(z)+∑4,T(z), where the random vector θT∗ is of the form (24).

Consider sample continuous Gaussian processes ηjT(z)=djT−1(θ0) ∫0Tε(t)gj(t+z,θ0)dt,z∈[0,H],T>0,j=1,q‾. 0,\hspace{2.5pt}\hspace{2.5pt}j=\overline{1,q}.\]]]>

For z1,z2∈[0,H] we have EηjT(z1)ηjT(z2)=djT−2(θ0) ∫0T ∫0TB(t−s)gj(t+z1,θ0)gj(s+z2,θ0)dtds==djT−2(θ0) ∫z1T+z1 ∫z2T+z2B(t−s+z2−z1)gj(t,θ0)gj(s,θ0)dtds, and the double integral in (30) can be symbolically written as ∫z1T+z1 ∫z2T+z2= ∫0T+ ∫TT+z1− ∫0z1 ∫0T+ ∫TT+z2− ∫0z2. Bound the integral djT−2(θ0) ∫0T ∫TT+z2≤ ∫TT+z2 ∫0TB2(t−s+z2−z1)dtds1/2××djT−2(θ0) ∫TT+z2gj2(s,θ0)ds1/2≤≤H1/2‖B‖2djT−2(θ0)dj,T+H2(θ0)−djT2(θ0)1/2⟶0,asT→∞, due to Lemma 2. Similarly djT−2(θ0)∫0T+z1∫0T⟶0 , as T→∞ . Also it is easy to see that djT−2(θ0) ∫0T ∫0z2≤H1/2‖B‖2djH(θ0)djT−1(θ0)⟶0,djT−2 ∫0z1 ∫0T⟶0,djT−2(θ0) ∫TT+z1 ∫TT+z2≤HB(0)djT−2(θ0) ∫TT+Hgj2(t,θ0)dt⟶0,djT−2(θ0) ∫0z1 ∫TT+z2≤HB(0)djH(θ0)djT−1(θ0)djT−2(θ0) ∫TT+Hgj2(s,θ0)ds1/2⟶0,djT−2(θ0) ∫0T+z1 ∫0z2⟶0,djT−2(θ0) ∫0z1 ∫0z2⟶0,asT→∞. Thus for any z1,z2∈[0,H] and j=1,q‾ . EηjT(z1)ηjT(z2)=BjT(z1−z2)+ojT(1),ojT(1)⟶0,asT→∞, and all finite-dimensional distributions of the Gaussian processes {ηjT(z),z∈[0,H]} , j=1,q‾ , converge, as T→∞ , to the corresponding finite-dimensional distributions of the stationary Gaussian processes ξj,j=1,q‾ , with covariance functions (25).

Besides, under conditions N1(ii) and R3 E(ηjT(z1)−ηjT(z2))2=djT−2(θ0) ∫0T ∫0TB(t−s)(gj(t+z1,θ0)−gj(t+z2,θ0))××gj(s+z1,θ0)−gj(s+z2,θ0)dtds≤≤djT−2(θ0) ∫0T ∫0T|B(t−s)|gj(t+z1,θ0)−gj(t+z2,θ0)2dtds≤≤‖B‖1djT−2(θ0)ΨjT(z1,z2;θ0)≤k‾j‖B‖1|z1−z2|2, z1,z2∈[0,H] , j=1,q‾ , ‖B‖1=∫R|B(t)|dt<∞ .

We have proved that ‖ηjT‖⟶D‖ξj‖ , j=1,q‾ , and from the condition AN it follows ∑3,T‖⟶P0,asT→∞.

Note further that similarly to (26) ∫0T|ε(t)|dt≤T2(1+sT∗) , and if the events AT(r) , BT∗=sT∗≤1+B(0) occur, then the norm of any term ∑4,Tij , of the sum ∑4,T can be dominated in the following way (compare with (27)): ∑4,Tij≤12(T+H)1/2di,T+H−1(θ0)dj,T+H−1(θ0)dij,T+H(θ0)××diT−1(θ0)di,T+H(θ0)djT−1(θ0)dj,T+H(θ0)××(T+H)1/2dij,T+H−1(θ0)supt∈[0,T+H],u∈Vc(r)∩UT(θ0)hij(t,u)××T(T+H)−1(1+sT∗)αiTT−1/2αjT≤≤12(1+β)2r2kij(r)k˜ij(r)(2+B(0))T−1/2 for any β>0 0$]]> and T>T0 {T_{0}}$]]>. In addition (compare with (29)), PBT∗‾≤2‖B‖22T−1, and from (28), (31), and (32) we obtain ‖∑4,T‖⟶P0,asT→∞.  □

Theorem 2 is proved as well.

In the proofs of the sections 3 and 4 the condition R2(i) has been used just for r=0 . However this condition is used for any r≥0 in the proof of LSE θˆT asymptotic normality: see explanation in the example below.

 Trigonometric regression function

In this section, we consider the example of trigonometric regression function g(t,θ0)= ∑k=1NAk0cosφk0t+Bk0sinφk0t, where θ0=(θ10,θ20,θ30,…,θ3N−20,θ3N−10,θ3N0)=A10,B10,φ10,…,AN0,BN0,φN0, Ck02=Ak02+Bk02>0 0$]]>, k=1,N‾ , φ0=φ10,…,φN0∈Φ(φ_,φ‾) ,

Φ(φ_,φ‾)=φ=(φ1,…,φN)∈RN:0≤φ_<φ1<⋯<φN<φ‾<+∞ .

To apply the results obtained in the paper to the function (33), we have to change a bit the Definition 1 of the LSE. We will use the following modification of LSE proposed by Walker [24], see also Ivanov [12, 13]. Consider non-decreasing system of open convex sets ST⊂Φ(φ_,φ‾) , T>T0>0 {T_{0}}>0$]]>, given by the condition that the true value of unknown parameter φ0∈ST , limT→∞ST=Φ(φ_,φ‾) , and limT→∞inf1≤j<k≤N,φ∈STT(φk−φj)=+∞,limT→∞infφ∈STTφ1=+∞.

The LSE in the Walker sense of unknown parameter (34) in the model (1) with regression function (33) is said to be any random vector θˆT=Aˆ1T,Bˆ1T,φˆ1T,…,AˆNT,BˆNT,φˆNT∈ΘTc having the property QT(θˆT)=minτ∈ΘTcQT(τ), where QT(τ) is defined in (2) and ΘT⊂R3N is such that Ak∈R , Bk∈R , k=1,N‾ , and φ∈ST .

The relations (35) allows to distinguish the parameters φk , k=1,N‾ , and prove the consistency of the LSE θˆT in the Walker sense, see [24, 12, 13], and [18].

Suppose the assumption (35) is satisfied for the LSE in the Walker sense of the parameters (33). Then under conditions N1 and N2 the relation (23) of Theorem 2 holds true.

Due to the smoothness of function (33) with respect to the totality of variables, there is no need to introduce conditions for the differentiability of the function g by the variables θ in the set Θγ and by the variable t in the set (−γ,+∞) , as it was done in the main part of the paper for technical necessity.

To check the fulfillment of the condition R1 for regression function (33) we get Akcosφkt+Bksinφkt−Ak0cosφk0t−Bk0sinφk0t≤≤Ak−Ak0+Bk−Bk0+Ak0+Bk0tφk−φk0,k=1,N‾, and therefore ΦT(θ,θ0)≤3N ∑k=0NTAk−Ak02+TBk−Bk02++13Ak0+Bk02T3φk−φk02. Note that for k=1,N‾ T−1d3k−2,T2(θ0),T−1d3k−1,T2(θ0)⟶12,T−3d3k,T2(θ0)⟶16Ck02,asT→∞. Thus for any ε>0 0$]]> and T>T0=T0(ε) {T_{0}}={T_{0}}(\varepsilon )$]]> from (38) it follows Td3k−2,T−2(θ0)<2+ε,Td3k−1,T−2(θ0)<2+ε,T3d3k,T−2(θ0)<6(Ck0)−2+ε. Increasing T0 , if necessary, we obtained from (37) and (39) ΦT(θ,θ0)≤3N ∑1N(2+ε)d3k−2,T2(θ0)Ak−Ak02++(2+ε)d3k−1,T2(θ0)Bk−Bk02++2Ak0+Bk02Ck02+εd3k,T2(θ0)φk−φk02. So, as it follows from (40), for any θ0∈Θ and ε>0 0$]]> there exists T0>0 0$]]> such that for T>T0 {T_{0}}$]]> the inequality (13) of the condition R1 is satisfied with constant k0≥12N+ε .

In the conditions R2(i) and R2(ii), instead of sets UT(θ0) , one should take sets U˜T(θ0)=dT(θ0)ΘTc−θ0 , and verification of conditions R2 for function (33) is routine.

Check condition R3 (see (12), (14)). Obviously, for k=1,N‾ g3k−2′(t,θ)=−φksinφkt,g3k−1′(t,θ)=φkcosφkt,g3k′(t,θ)=−Aksinφkt−Aktφksinφkt+Bkcosφkt−Bktφksinφkt. Let z1<z2 , then g3k(t+z1,θ)−g3k(t+z2,θ)=g3k′(t∗,θ)|z1−z2|, where t∗=t+z1+ν(z2−z1)≤t+H , ν∈(0,1) is some number.

Using formula (41), we obtain g3k′(t∗,θ0)2≤2Ak02sinφk0t∗+φk0t∗sinφk0t∗2++2Bk02cosφk0t∗−φk0t∗sinφk0t∗2≤2Ck021+φ‾(H+t)2. From (39) and (42) it follows for any ε>0 0$]]>, θ0∈Θ , and T>T0 {T_{0}}$]]> the inequalities (14) are correct with constants k‾i≥4φ‾2+ε , if i=3k , k=1,N‾ . Similarly we obtain the inequalities (14) with constants k‾i≥2φ‾2+ε , if i=3k−2,3k−1 , k=1,N‾ .

Passing to condition R4, we note that the spectral measures of the trigonometric regression were studied by Whittle [25], Walker [24], Hannan [7], Ivanov [12], Ivanov et al. [18]. For regression function (33) spectral measure μ(dλ;θ0) , θ0∈Θ , is a block-diagonal matrix diagMk(θ0),k=1,N‾ , where Mk(θ0)=δkiρkβ‾k−iρkδkγ‾kβkγkδk,βk=32Ck0(Bk0δk+iAk0ρk),γk=32Ck0(−Ak0δk+iBk0ρk), with δk=δk(dλ) , and the signed measure ρk=ρk(dλ) being located at the points ±φk0 , k=1,N‾ . Moreover, δk±φk0=12 , ρk±φk=±12 , k=1,N‾ . On the other hand, μT(R;θ0)= ∫−∞∞μ(dλ;θ0)=J(θ0)=diagJk(θ0),k=1,N‾,Jk(θ0)=1032Bk0(Ck0)−101−32Ak0(Ck0)−132Bk0(Ck0)−1−32Ak0(Ck0)−11.

Since detJk=14 , the matrix (44) is positive definite. Practically the components of the matrix-valued measure μ(dλ;θ)=μjl(dλ;θ)j,l=1q , q=3N in our example, are determined from relations Rjl(h,θ)=limT→∞djT−1(θ0)dlT−1(θ0) ∫0Tgj(t+h,θ)gl(t,θ)dt==∫Reiλhμjl(dλ;θ),h∈R, where it is supposed that the matrix function Rjl(h;θ)j,l=1q is continuous at h=0 .

As to the condition AN fulfillment for trigonometric regression function (33) in the paper Ivanov et al. [18], it is shown using relations (38) that normalized LSE in the Walker sense T1/2Aˆ1T−A10,T1/2Bˆ1T−B10,T3/2φˆ1T−φ10,…,T1/2AˆNT−AN0,T1/2BˆNT−BN0,T3/2φˆNT−φN0 is asymptotically, as T→∞ , normal N0,∑TRIG , where ∑TRIG is a block-diagonal matrix with blocks 4πf(φk0)(Ck0)2(Ak0)2+4(Bk0)2−3Ak0Bk0−6Bk0−3Ak0Bk0(Ak0)2+4(Bk0)26Ak0−6Bk06Ak0(Ak0)2+4(Bk0)2,k=1,N‾.

To obtain such a result, it was first proved in [18] that the normalized estimator (36) is weakly consistent, that is, for any r>0 0$]]> PT−1/2dT(θ0)(θˆT−θ0)≥r⟶0,asT→∞. Then, under a complex set of conditions for general regression function asymptotic normality of the LSE of its parameters was proved. And finally, it was verified that the trigonometric regression function satisfies the specified set of conditions. It is important to note that the proofs of the asymptotic normality of the LSE complying with Definition 1 and Definition 5 are the same.

It remains to check the last condition RN associated with the regression function (33). As mentioned above under assumptions N1 and N3 the condition RN follows from N1, R4. If the function b(λ) , λ∈R , is not bounded, then we verify the convergence (18) using Lemma 3.

First of all for θ0∈Θ in view of (43) ∫R|λ|1+δf(λ)μ3k−i,3k−i(dλ;θ0)=(φk0)1+δf(φk0)<∞,i=0,1,2,k=1,N‾, and RN1(iii) is true. Suppose that the condition RN(ii) is also satisfied. It can, for example, happen when outside of some neighborhood of zero the spectral density f(λ),λ∈R , will behave as a function C|λ|lna(1+|λ|),a>1 1$]]>.

Using formulas (41) and the fact that it can be taken, for example, |λ|>φ‾+1=λ0 \overline{\varphi }+1={\lambda _{0}}$]]> in calculating the integrals (19), there are no non-integrable singularities of the form 1λ±φk0 , k=1,N‾ . Moreover, in the considered example a sharpened version of inequalities (20) of the condition RN(i) holds: sup|λ|>λ0djT−2(θ0)g˜Tj(λ,θ0)2≤hj(θ0)T−1,j=1,3N‾,θ0∈Θ. {\lambda _{0}}}{\sup }{d_{jT}^{-2}}({\theta ^{0}}){\left|{\widetilde{g}_{T}^{j}}(\lambda ,{\theta ^{0}})\right|^{2}}\le {h_{j}}({\theta ^{0}}){T^{-1}},\hspace{2.5pt}j=\overline{1,3N},\hspace{2.5pt}{\theta ^{0}}\in \Theta .\]]]>  □

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