Asymptotics for functionals of powers of a periodogram

We present large sample properties and conditions for asymptotic normality of linear functionals of powers of the periodogram constructed with the use of tapered data.

In this paper, we will study large sample properties of the empirical spectral functionals of the form for appropriate functions ϕ(λ) with ϕ(λ)f k 2 (λ) ∈ L 1 (Λ), where I k T (λ) is the kth power of the periodogram based on the tapered data {h T (t)Y (t), t ∈ K T }, and k is a positive integer. The taper will be of the form h T (t) = h( t T ) with h satisfying the following standard assumption.
Assumption H. The function h(t), t ∈ R, is an even positive function of bounded variation with bounded support: h(t) = 0 for |t| > 1.
The periodogram corresponding to the tapered data is defined as where d T (λ) is the finite Fourier transform based on tapered data: and we suppose that H 2,T (0) = 0. Functionals of the form (1) for k = 1 have been extensively studied in the literature, in particular, due to their applications for parameter estimation in the spectral domain: their behavior as T → ∞ is important for establishing asymptotic properties of so-called minimum contrast estimators such as Whittle and Ibragimov estimators (see, e.g., [9,1], and references therein). The case of the squared periodogram was treated, for example, in [8], with application to a goodness-of-fit statistics, and in [12], with application to minimum contrast estimation.
Asymptotic results for the functionals of the form (1) with general k ≥ 2 were studied in [6] and applied to derivation of properties of weighted least squares estimators in the frequency domain and also in [11], with several applications discussed, in particular, a frequency domain goodness-of-fit testing.
In this paper, we derive asymptotic results for functionals (1) with general k ≥ 2 in a more general setting, using the tapered data, and under a different set of conditions in comparison with those used in [6] and [11]; in the Gaussian case, we state our results in terms of integrability conditions for the spectral density and weight function. Methods for the proofs are similar to those used in [12] with appropriate modifications required for the more general case under consideration in the present paper.

Results and discussion
We begin with the following assumptions.

Assumption 2.
The spectral densities f k (λ 1 , . . . , λ k−1 ), k = 2, 3, . . . , of the stochastic process Y (t) are bounded and continuous. In what follows, we denote the second-order spectral density f 2 (λ) simply by f (λ) omitting the subscript 2. Theorem 1. Let Assumptions 1, 2, and H hold, and let the functions ϕ, ϕ 1 (λ), . . . , ϕ k (λ) satisfy Assumption 3. Then, as T → ∞, Suppose now that the process Y (t) is Gaussian. In this case the above asymptotic results can be stated under the conditions of integrability on the weight function and spectral density.
(1) If p and q satisfy the relation then, as T → ∞, (2) If p and q satisfy the relation (3) If p and q satisfy the relation then the cumulants of orders r ≥ 3 of the normalized functionals J k,T (ϕ i ) tend to zero as T → ∞: cum T 1/2 J k,T (ϕ 1 ), . . . , T 1/2 J k,T (ϕ r ) → 0.
As corollaries of the above theorems, we obtain the next asymptotic normality results.
Let us fix the weight functions ϕ 1 , . . . , ϕ m and denote ..,m be a Gaussian random vector with zero mean and secondorder moments Assumption 4. The spectral density of the second order f (λ), the weight function ϕ(λ), and the taper h are such that T 1/2 (EJ k,T (ϕ) −J k (ϕ)) → 0.
moreover, if Assumption 4 holds for the functions ϕ i , i = 1, . . . , m, then ..,m be a Gaussian random vector with zero mean and secondorder moments , t ∈ Z, be a Gaussian stationary process with spectral density f (λ) ∈ L p (Λ), and let the functions ϕ 1 , . . . , ϕ m ∈ L q (Λ), where 1 ≤ p, q ≤ ∞, be such that 1 q + k 1 p = 1 2 . Suppose also that Assumption H holds. Then moreover, if Assumption 4 holds for the functions ϕ i , i = 1, . . . , m, then We next state more general results, the joint asymptotic normality for functionals of different powers of the periodogram. Consider ..,m be a Gaussian random vector with zero mean and secondorder moments ..,m be a Gaussian random vector with zero mean and secondorder moments Theorem 6. Let Y (t), t ∈ Z, be a Gaussian stationary process with a spectral density f (λ) ∈ L p (Λ), and let the functions ϕ 1 , . . . , ϕ m ∈ L q (Λ), where 1 ≤ p, q ≤ ∞, be such that 1 q + min{k i } 1 p = 1 2 . Suppose also that Assumption H holds. Then

Remark 1.
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.

Remark 2.
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.

Remark 3.
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.

Proofs
For the proofs of the results of Section 2, we use the technique based the properties of the multidimensional kernels of Fejér type (see, e.g., [5,1], and references therein) and the Hölder-Young-Brascamp-Lieb inequality (see [3,4], and references therein; see also [12]). In what follows, we will refer to the latter as the HYBL inequality. The application of these tools leads to very transparent and elegant proofs. We will also use the formula giving expressions for cumulants of products of random variables via products of cumulants of the individual variables (see, e.g., [10,5]) and the multilinearity property of cumulants. The lines of reasonings are very close to those used in [12].

Proof of Theorem 1. Consider
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 λ i with odd indices are simply equal to λ, whereas λ i with even i are equal to −λ. Then we can write down the expectation in the following form: The cumulants of the finite Fourier transforms d T (λ), λ ∈ Λ, can be written as follows: Correspondingly, we obtain the following formula for the expectation of J k,T (ϕ): 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 = (γ 1 , . . . , γ 2k ) due to p restrictions on the variables γ j , j = 1, . . . , 2k, described by the Kronecker delta functions δ. Now we note that the products k j=1 H 1,T (λ j ) in the case where k j=1 λ j = 0 give rise to a class of δ-type kernels (or Fejér-type kernels). Namely, if Assumption H holds and H k,T (0) = 0, then is a kernel over Λ k−1 , which is an approximate identity for convolution (see, e.g., [7]), and provided that the function G(·, . . . , ·) is bounded and continuous at the point (v 1 , . . . , v k−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(d T (λ), d T (−λ)) appear in (2), we obtain under the integral sign in (3) the terms of the form We note that there are k! such terms, therefore, in the expression for EJ k,T (ϕ), we have the term and this is the only case where we have k kernels, and all k factors 1 2πH2,T (0) are used to compose these kernels Φ h 2,T (·). In all other partitions, we will be able to compose from 1 to k − 1 kernels taking combination of H 1,T (·) with suitable arguments: for each 2nd-order kernel, we will use one of the factors 1 2πH2,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−1 H l,T (0) , and therefore we will modify the factor 1 (2πH2,T (0)) k by taking, instead, (2π) l−1 H l,T (0) (2πH2,T (0)) k . So, for those partitions, when we compose kernels of orders, say, l 1 , . . . , l r , with r i=1 l i = 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 l 1 , . . . , l r , and the factor r i=1 (2π) li−1 H li,T (0) (2πH 2,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: Now we take into account the following asymptotics for H k,T (0): H k,T (0) ∼ T H k (0), where H k (0) = h k (λ) dλ, and conclude that, in the case of r kernels, 1 ≤ r ≤ k − 1, the factor (7) is asymptotically of order 1 T k−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 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 where the summation is taken over all indecomposable partitions ν = (ν 1 , . . . , ν p ), |ν i | > 1, of the table T 2 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, where µ i , λ i ∈ {α, −α, β, −β}, and ν = {(µ i , λ i ), i = 1, . . . , k + l} forms an indecomposable partition of the table T 2 .
In the case where we have the k − 1 cumulants cum(d T (α), d T (−α)) and l − 1 cumulants cum(d T (β), d T (−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 1 T . Only the terms of this kind in (9) give the main contribution (of order 1 T ) 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: Their contribution to the covariance is of the form Taking into account formula (5), we can evaluate the latter as 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 in the sum (9) (there are klk!l! such terms), and the corresponding contribution to the covariance (8) is of the form 2 ϕ 1 (α)ϕ 2 (β)f (α)f (β)f 4 (α, −α, β) dα dβ as T → ∞.
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 1 T 2 . Summarizing the above reasonings, we come to the asymptotics for the covariance as given in statement (2) of the theorem.
If we choose r = 2 and take into account that, under Assumption H, we have H 1,T r ≤ CT 1−1/r and H 1,T (0) ∼ T , then from (14) we arrive at the bound const ϕ q f k p as T → ∞, with the restrictions on p and q as in statement (1) of the theorem. From this point we can repeat the same arguments as in [12] to show that, in fact, this bound can be strengthen to o(1) as T → ∞.