E|Y(t)|k<∞ for all k, and Y(t) has (cumulant) spectral densities of orders k=2,3,… , that is, there exist the functions fk(λ1,…,λk−1)∈L1(Λk−1) , Λ=(−π,π] , k=2,3,… , such that the cumulant function of order k is given by ck(t1,…,tk−1)=∫Λk−1fk(λ1,…,λk−1)eiΣ1k−1λjtjdλ1…dλk−1.

The function h(t) , t∈R , is an even positive function of bounded variation with bounded support: h(t)=0 for |t|>1 1$]]>.

The spectral densities fk(λ1,…,λk−1) , k=2,3,… , of the stochastic process Y(t) are bounded and continuous.

The weight function φ(λ) is bounded and continuous.

Let Assumptions 1, 2, and H hold, and let the functions φ,φ1(λ),…,φk(λ) satisfy Assumption 3. Then, as T→∞ , (1)

EJk,T(φ)→k!∫Λφ(λ)fk(λ)dλ;

Let Y(t) , t∈Z , be a Gaussian stationary process with spectral density f(λ) , λ∈Λ , such that f(λ)∈Lp(Λ) , and let the functions φ,φ1,…,φk∈Lq(Λ) , where 1≤p,q≤∞ . Suppose also that Assumption H holds. (1)

If p and q satisfy the relation 1q+k1p=1, then, as T→∞ , EJk,T(φ)→k!∫Λφ(λ)fk(λ)dλ.

The spectral density of the second order f(λ) , the weight function φ(λ) , and the taper h are such that T1/2(EJk,T(φ)−Jk˜(φ))→0 .

Let Assumptions 1, 2, and H hold, and let the functions φi , i=1,…,m , satisfy Assumption 3. Then T1/2(JT−EJT)→Dξas T→∞; moreover, if Assumption 4 holds for the functions φi , i=1,…,m , then T1/2(JT−J˜)→Dξas T→∞.

Let Y(t) , t∈Z , be a Gaussian stationary process with spectral density f(λ)∈Lp(Λ) , and let the functions φ1,…,φm∈Lq(Λ) , where 1≤p,q≤∞ , be such that 1q+k1p=12 . Suppose also that Assumption H holds. Then T1/2(JT−EJT)→Dζas T→∞; moreover, if Assumption 4 holds for the functions φi , i=1,…,m , then T1/2(JT−J˜)→Dζas T→∞.

Let Assumptions 1, 2, and H hold, and let the functions φi , i=1,…,m , satisfy Assumption 3. Then T1/2(Jk1,…,km,T−EJk1,…,km,T)→Dξas T→∞; moreover, if Assumption 4 holds with k=ki , φ=φi , i=1,…,m , then T1/2(Jk1,…,km,T−J˜k1,…,km)→Dξas T→∞.

Integrals of nonlinear functions of the periodogram (including, in particular, powers of positive orders of the periodogram) were studied, for example, in [13] for discrete time processes under the assumption of boundedness of the spectral density. In [8], the integral functionals of the squared periodogram were studied for stationary Gaussian series given by the moving-average representation, and the asymptotic normality result was stated under the particular assumption of summability of the coefficients of the representation and continuity of the derivative of the spectral density. In [6] and [11], the asymptotic results for functionals of powers of the periodogram of general order have been studied under the conditions of summability of cumulants of the process. In this paper, we state the results for discrete-time non-Gaussian processes under the condition of boundedness of spectral densities of all orders (which are supposed to exist), and we also derive the results for Gaussian case under the conditions of integrability of the spectral density and weight function.

Conditions on the spectral density under which Assumption 4 will be satisfied can be formulated analogously to the corresponding conditions in [1] for the case where h(t)≡1 and analogously to the conditions in [2] for the general h(t) of Assumption H.

The results on asymptotic properties of the integrals of the powers of the periodogram can be useful for some problems of statistical inference. One possible application is hypothesis testing concerning the form of the spectral density of the process. For example, in [8], a quadratic goodness-of-fit test in the spectral domain was studied for Gaussian processes. Note that the asymptotic normality result for the corresponding test statistic stated in [8] can be also derived from our Theorem 6, that is, under a different set of conditions. More applications of the integrals of the powers of the periodogram for goodness-of-fit testing, peak testing, and assessing model misspecification are presented in [11]. In [6] and [12], the integrals of the squared periodogram were applied for parametric estimation in the spectral domain.

Consider EITk(λ)=1(2πH2,T(0))kE[(cum(dT(λ)dT(−λ))k]=1(2πH2,T(0))kE[cum(dT(λ)dT(−λ)⋯cum(dT(λ)dT(−λ)]. We apply now the formula for cumulants of products of random variables (see, e.g., [10]); it is convenient to assign the indices to λs in the following way: we can enumerate all λs appearing in the above row from 1 to 2k , having in mind that λi with odd indices are simply equal to λ, whereas λi with even i are equal to −λ . Then we can write down the expectation in the following form: (2) EITk(λ)=1(2πH2,T(0))k∑ν=(ν1,…,ν1)partition of (1,…,2k) ∏l=1pcum(dT(λi),i∈νl)× ∏i=1kδ(λ2i−1−λ) ∏i=1kδ(λ2i+λ).

The cumulants of the finite Fourier transforms dT(λ) , λ∈Λ , can be written as follows: cum(dT(α1),…,dT(αk))=∫KTk ∏i=1khT(ti)e−iΣ1kαjtjck(t1−tk,…,tk−1−tk)dt1…dtk=∫Λk−1fk(γ1,…,γk−1)× ∏j=1k−1H1,T(γj−αj)H1,T(− ∑j=1k−1γj−αk)dγ1…dγk−1, where H1,T(λ)=∫KThT(t)e−itλdt.

Correspondingly, we obtain the following formula for the expectation of  Jk,T(φ) : (3) EJk,T(φ)=E∫Λφ(λ)ITk(λ)dλ=∫Λφ(λ)1(2πH2,T(0))k∑ν=(ν1,…,νp)partition of (1,…,2k)∫Λ2k−p ∏i=1pf|νi|(γj,j∈ν˜i)× ∏j=12kH1,T(γj−λj) ∏l=1pδ(∑j∈νlγj) ∏i=1kδ(λ2i−1−λ) ∏i=1kδ(λ2i+λ)dγ′dλ.

Here and in similar formulas below, we use the following notation: for a set of natural numbers ν, we denote by |ν| the number of elements in ν and by ν˜ the subset of ν that contains all elements of ν except the last one. Integration in the inner integral in the above formula is understood with respect to (2k−p) -dimensional vector γ′ obtained from the vector γ=(γ1,…,γ2k) due to p restrictions on the variables γj , j=1,…,2k , described by the Kronecker delta functions δ.

Now we note that the products ∏j=1kH1,T(λj) in the case where ∑j=1kλj=0 give rise to a class of δ-type kernels (or Fejér-type kernels). Namely, if Assumption H holds and Hk,T(0)≠0 , then (4) Φk,Th(λ1,…,λk−1):=1(2π)k−1Hk,T(0) ∏j=1k−1H1,T(λj)H1,T(− ∑j=1k−1λj) is a kernel over Λk−1 , which is an approximate identity for convolution (see, e.g., [7]), and (5) limT→∞∫Λk−1G(u1−v1,…,uk−1−vk−1)Φk,Th(u1,…,uk−1)du1…duk−1=G(v1,…,vk−1), provided that the function G(·,…,·) is bounded and continuous at the point (v1,…,vk−1) .

The asymptotic behavior of the right-hand side of (3) can be evaluated basing on the property (5).

Let us first consider the partitions ν composed by pairs. For those partitions, when the products of only the cumulants of the form cum(dT(λ),dT(−λ)) appear in (2), we obtain under the integral sign in (3) the terms of the form 1(2πH2,T(0))k{∫f(γ)H1,T(γ−λ)H1,T(−γ+λ)dγ}k={∫f(γ)Φ2,Th(γ−λ)dγ}k. We note that there are k! such terms, therefore, in the expression for EJk,T(φ) , we have the term (6) k!∫Λφ(λ){∫Λf(γ)Φ2,Th(γ−λ)dγ}kdλ, and this is the only case where we have k kernels, and all k factors 12πH2,T(0) are used to compose these kernels Φ2,Th(·) .

In all other partitions, we will be able to compose from 1 to k−1 kernels taking combination of H1,T(·) with suitable arguments: for each 2nd-order kernel, we will use one of the factors 12πH2,T(0) from 1(2πH2,T(0))k ; otherwise, when composing the lth order kernel with l≠2 , we will need the normalizing factor 1(2π)l−1Hl,T(0) , and therefore we will modify the factor 1(2πH2,T(0))k by taking, instead, (2π)l−1Hl,T(0)(2πH2,T(0))k .

So, for those partitions, when we compose kernels of orders, say, l1,…,lr , with ∑i=1rli=2k , the corresponding integral in (3) will be represented in the form of a generalized convolution of some product of spectral densities of different orders with the product of kernels of orders l1,…,lr , and the factor (7) ∏i=1r(2π)li−1Hli,T(0)(2πH2,T(0))k will be supplied to the integral. For example, in the simplest case where p=1 , the corresponding term in (3) can be represented as follows: (2π)2k−1H2k,T(0)(2πH2,T(0))k∫Λφ(λ)∫Λ2k−1f2k(γ1,…,γk−1)×Φ2k,Th(γ1−λ1,…,γ2k−1−λ2k−1)× ∏i=1kδ(λ2i−1−λ) ∏i=1kδ(λ2i+λ) ∏i=12k−1dγidλ.

Now we take into account the following asymptotics for Hk,T(0) : Hk,T(0)∼THk(0) , where Hk(0)=∫hk(λ)dλ , and conclude that, in the case of r kernels, 1≤r≤k−1 , the factor (7) is asymptotically of order 1Tk−r ; the corresponding integrals containing these kernels will converge to finite limits under the conditions of the theorem according to (5). Therefore, the expectation is obtained as the limit of (6). This gives statement (1) of the theorem.

Consider (8) cov(Jk,T(φ1),Jl,T(φ2))=1(2πH2,T(0))k+l×∫Λ2φ1(α)φ2‾(β)cum((dT(α)dT(−α))k,(dT(β)dT(−β))l)dαdβ. The cumulant under the integral sign in (8) according to the formula for calculation of cumulants of products of random variables can be written in the form (9) ∑ν=(ν1,…,νp) ∏i=1pcum(dT(μj),μj∈νi), where the summation is taken over all indecomposable partitions ν=(ν1,…,νp) , |νi|>1 1$]]>, of the table T2 with two rows, {α,−α,…,α,−α} (of length 2k ) and {β,−β,…,β,−β} (of length 2l ). For asymptotic analysis of expression (8), we can use the reasonings analogous to those for the case of functionals of squared periodogram in [12], but now, dealing with the tapered case, we need to keep track of normalizing factors for appearing kernels. Again, similarly to the previous consideration of the expectation, we analyze all possible partitions and kernels that can be composed for every particular partition. Let us first consider the terms in (9) that correspond to partitions by pairs, that is, (10) ∏i=1k+lcum(dT(μi),dT(λi)), where μi,λi∈{α,−α,β,−β} , and ν={(μi,λi),i=1,…,k+l} forms an indecomposable partition of the table T2 .

In the case where we have the k−1 cumulants cum(dT(α),dT(−α)) and l−1 cumulants cum(dT(β),dT(−β)) in the product (10), according to formula (4), we can compose k+l−1 kernels ( k+l−2 kernels of order 2 and one of order 4), and the factor before the integral in (8) becomes of the form 2πH4,T(0)(H2,T(0))2 , which is asymptotically of order 1T . Only the terms of this kind in (9) give the main contribution (of order 1T ) into the covariance (8); all other terms in (10) produce a smaller-order contribution to the covariance (8). More precisely, in order to describe the asymptotics of the covariance, we have to consider, among the terms in (9), the following ones: (11) (cum(dT(α),dT(−α)))k−1(cum(dT(β),dT(−β)))l−1×[cum(dT(α),dT(β))cum(dT(−α),dT(−β))+cum(dT(α),dT(−β))cum(dT(−α),dT(β))]. Their contribution to the covariance is of the form 1(2πH2,T(0))k+l∫∫φ1(α)φ2‾(β)×[∫f(γ1)H1,T(γ1−α)H1,T(−γ1+α)dγ1]k−1×[∫f(γ2)H1,T(γ2−β)H1,T(−γ2+β)dγ2]l−1×[∫f(γ3)H1,T(γ3−α)H1,T(−γ3−β)dγ3×∫f(γ4)H1,T(γ4+α)H1,T(−γ4+β)dγ4+∫f(γ3)H1,T(γ3−α)H1,T(−γ3+β)dγ3×∫f(γ4)H1,T(γ4+α)H1,T(−γ4−β)dγ4]dαdβ=2πH4,T(0)(H2,T(0))2∫∫φ1(α)φ2‾(β)[∫f(γ1)Φ2,Th(γ1−α)dγ1]k−1×[∫f(γ2)Φ2,Th(γ2−β)dγ2]l−1×∫∫f(γ3)f(γ4)[Φ4,Th(γ3−α,−γ3−β,γ4+α)+Φ4,Th(γ3−α,−γ3+β,γ4+α)]dγ3dγ4dαdβ. Taking into account formula (5), we can evaluate the latter as ∼2πT∫h4(t)dt(∫h2(t)dt)2∫Λφ1(λ)[φ2‾(λ)+φ2‾(−λ)]fk+l(λ)dλasT→∞. We note also that there are klk!l! terms of the form (11) in (9).

Let us now consider the terms in (9) that correspond to other partitions. We can see that one more possibility is left to compose k+l−1 kernels, namely, the case of the terms of the form [cum(dT(α),dT(−α))]k−1[cum(dT(β),dT(−β))]l−1×cum(dT(α),dT(−α),dT(β),dT(−β)) in the sum (9) (there are klk!l! such terms), and the corresponding contribution to the covariance (8) is of the form 1(2πH2,T(0))k+l∫∫φ1(α)φ2‾(β)×[∫f(γ1)H1,T(γ1−α)H1,T(−γ1+α)dγ1]k−1×[∫f(γ2)H1,T(γ2−β)H1,T(−γ2+β)dγ2]l−1×∫∫∫f4(μ1,μ2,μ3)H1,T(μ1−α)H1,T(μ2+α)×H1,T(μ3−β)H1T(− ∑i=13μi+β)dμ1dμ2dμ3dαdβ=2πH4,T(0)(H2,T(0))2∫∫φ1(α)φ2‾(β)×∫f(γ1)Φ2,Th(γ1−α)dγ1∫f(γ2)Φ2,Th(γ2−β)dγ2×∫∫∫f4(μ1,μ2,μ3)Φ4,Th(μ1−α,μ2+α,μ3−β)dμ1dμ2dμ3dαdβ∼2πT∫h4(t)dt(∫h2(t)dt)2∫∫φ1(α)φ2‾(β)f(α)f(β)f4(α,−α,β)dαdβasT→∞. In all other cases, we can compose less than k+l−1 kernels; the corresponding integrals will converge to finite limits supplied by the factor of orders not exceeding  1T2 .