Exploring spectral density estimation for spatial linear process with mixing innovations

We study the asymptotic properties of the spectral density estimator (a periodogram) of a linear spatial process with alpha mixing innovations. A periodogram is a natural estimate of the spectral density. Under some conditions, a relation between the periodograms of innovations and that of the linear process is established in a spatial case. As the estimator of periodogram is inconsistent, a linear filter is introduced and convergence properties of the obtained smoother periodogram estimator are studied.

to deal with this type of problem, see for examples [2,3,8,13], and most recently, [11]. A discussion on several models is done to deal with analysis of spatial data [16].
In this work, we are interested by the properties of spatial linear processes with alpha mixing innovations and mainly on periodogram. The lack of spatial asymptotic results of the periodogram of spectral density of these processes motivates this work. Hence, we study the asymptotic properties of this periodogram. A periodogram is a natural estimate of the spectral density. Under some conditions, a relation between the periodogram of innovations and the periodogram of linear process is established in a spatial case. As the estimator of periodogram is inconsistent, a linear filter is introduced and convergence properties of the obtained smoother periodogram estimator are studied.
The periodogram in the non-spatial case (N = 1) has been studied by [6], when the innovations are independent while [5] has studied the case where innovations are strongly mixing. We extend this last work to random fields (N > 1). Some recent papers deal with spatial periodogram smoothing but there is no work concerning weak processes. Different filters are used in literature, see, for example, [19] which focuses on kernel estimator of spectral density, with optimal smoothing number estimated from the data. The author studied consistency and asymptotic distribution of this estimator, with an automatic estimate of this smoothing number. In [9], the authors have studied asymptotic properties of smoothed non-parametric kernel spectral density estimator in the case of continuous stationary spatial process under shrinking asymptotic framework. This process is mixing but not necessarily derived from weak process as in the present work. They considered the bias and variances terms for tapered and untapered estimators, and obtained optimal bandwidth which minimizes the average mean squared error.
Let Z N , N ≥ 1, denote the integer lattice points in the N -dimensional Euclidean space. Let the process (X t , t ∈ Z N ) be a stationary spatial linear process defined on a probability space (Ω, A, P), of unknown spectral density f X : X t = s∈Z N a s Z t−s , t ∈ Z N , where the (Z t ) is a strictly stationary process with unknown spectral density f Z . Z t are real-valued α mixing, and identically distributed and uncorrelated random variables with zero mean and finite second moment σ 2 . The spectral density of (X t ) is given by where T is the transpose operator and γ (k) = Cov(X k+u , X u ) is the auto-covariance of (X t , t ∈ Z N ). A point k in Z N will be referred to as a site and written as k = k 1 , . . . , k N . Let n = n 1 , . . . , n N , we write n → ∞ if min {n k } → ∞ and 1 < j, k < N . Define n = n 1 · · · n N as sample size. All limits are taken as n → ∞. (X t , t ∈ Z N ) is observed over the rectangular domain In what follows, let i ≤ j denote ∃k, i k ≤ j k , i l = j l , l = 1, . . . , k − 1 (the lexicographic order). A simple natural estimator of f X (ω j ) appears to be If γ (.) is an absolutely summable auto-covariance function of the process (X t , t ∈ Z N ), then f X is continuous and the definition (2) is equivalent to where S = {k; k j = 1 − n j , . . . , n j − 1, j = 1, . . . , N } and with mean X = 1 n t∈I n X t and where I (k) = {t ∈ I n , t + k ∈ I n }. Hence the natural spectral density estimator is given by As in the non-spatial case (N = 1), this estimator is unbiased but not consistent. The consistent estimator is obtained by applying a linear filter to smooth the periodogram.
In this paper, we will deal with asymptotic properties of spectral density estimates f X of (X t, t ∈ Z N ) using observations on I n and the following conditions: Mixing conditions: Consider the α-mixing coefficient of the field (Z t , t ∈ Z N ). The spatial dependence of the process will be measured by this α mixing defined by where, for B and C, two σ -fields of A: α (B, C) = sup B∈B,C∈C |P(B ∩ C) − P(B)P(C)|. In the following, we suppose that the process is geometrically strongly mixing (GSM), that is, there exist β > 0 and a constant C > 0 such that This paper is organized as follows. In Sect. 2, we deal with asymptotic properties of the spatial periodograms of (Z t ) and (X t ). In Theorem 1, we first give a result that concerns an asymptotic distribution of the periodogram of innovations. Then in Theorem 2, we establish a relation between the periodogram of the innovations and that of (X t ); hence, the convergence of the periodogram of (X t ) to exponential independent variables is given in Theorem 3. In Sect. 3, we study the convergence of a smoothed periodogram of (X t ). Finally, Sect. 4 is devoted to some numerical results of the proposed periodogram.

Convergence of the periodogram
To estimate f X at arbitrary non-zero frequencies in the interval [−π, π] N , we need to extend the domain of I n,X to the whole interval [−π, π] N . The periodogram is then defined as a piecewise constant function which coincides with (2) in frequency ω j . For any ω ∈ [−π, π] N , the periodogram is defined as follows (using (2)): Let g(n, ω) = (g (n 1 , ω) , . . . , g(n N , ω)) be the multiple of ( 2π n 1 , . . . , 2π n N ) closest to ω. And let g(n, ω) = g(n, −ω) for ω ∈ [−π, 0] N . Then I n,X (ω) = I n,X (g(n, ω)). Set, for Then I n,X (ω j ) = To establish the Theorems 1 and 2 below, we need the following assumptions: Assumption 2 Assume that the mixing coefficient of (Z t ) satisfies These two assumptions hold in such problems and under our conditions on mixing process (geometric). They are used to manipulate our spatial inequalities as in Gao [10]. Let I n,Z be the periodogram of (Z t ) obtained by replacing (X t ) by (Z t ) in (6). We begin by a consistency result of I n,Z for all t.
Proof For an arbitrary λ ∈ [0, π] N , define A(λ) = A(g(n, λ)) and B(λ) = B(g(n, λ)), with Z t replacing X t ; since it suffices to show that converges in distribution to a centered Gaussian random vector with covariance matrix σ 2 I 2m . I 2m is the 2m × 2m identity matrix. Let u 1 , . . . , u m and v 1 , . . . , v m ∈ R, be fixed. Then set the random variables and To use the Cramer-Wold Theorem, we need the linear combination of the coordinates of (8). Assume for some integers r 1 , . . . , r N , we have n 1 = r 1 ( p + q), . . . , n N = r N ( p + q), q < p. The method of proof consists to define following large and small blocks used in [18]. Let, And so on. Note that Finally, Clearly, T (n, x, 1) is the sum of random variables in large blocks and T (n, x, i) for 2 ≤ i ≤ 2 N the sum for random variables in small blocks. If it is not the case that n 1 = r 1 ( p + q), . . . , n N = r N ( p + q) for some integers r 1 , . . . , r N , the term T (n, x, 2 N + 1) can be added. This term will not change the proof much (see [18]).
Consider T (n, x, 1), we enumerate in the arbitrary way the r = r 1 × · · · × r N terms U (1, n, x, j) of sum of T (n, x, 1) which we denote W 1 , . . . , W r . Note that the U (1, n, x, j) is measurable with respect to the σ -field generated by Y t , with t such that j k ( p + q) + 1 ≤ t k ≤ j k ( p + q) + p. Then distinct sets of sites I (1, n, x, j) = {t / j k ( p + q) + 1 ≤ t k ≤ j k ( p + q) + p} are far apart by distance of at least q and it contains p N sites. And note that r = n( p + q) −N ≤ n p −N .
We have According to Lemma 4.4 in [7], there exist independent random variables W * 1 , . . . , W * r such that for all m = 1, . . . , r We have Therefore, according to Markov inequality, we have Hence from Eq. (12), If we consider now It follows that . Now, we are going to prove the asymptotic normality of r p N −1 2 W * m . First, we verify following Lyapounouv conditions. That is, for some ρ > 2, we verify that We have We recall that We will now focus our attention on the numerator of Π , that is, Hence from Assumption 2 Since ρ > 2 and κ > 0, hence T (n, x, 1) has been treated and without loss of generality, consider T (n, x, 2). Enumerate the random variables U (2, n, x, j) in the arbitrary manner and refer to them as W 1 , . . . , W r . Now show that Since Z t is stationary, we have However, Note that as before Hence, We have n, x, j) is sum of Y t in the sites I (2, n, x, j). If j and j are in two distinct sites, then j k = j k for some 1 ≤ k ≤ N and j − j > q, since p > q. We obtain converges in probability to zero since q < p, hence the result.
(ii) In the following, we prove the result with λ instead of λ j and we make the substitution at the end of the proof. By the definition of I n,Z (λ), we have Hence E(I n,Z (λ)I n,Z (λ )) = n −2 i∈I n j∈I n s∈I n u∈I n Suppose that E(Z 4 i ) = θσ 4 and i∈I n α( i ) We will examine many cases: 1. i = j = s = u, Applying Davydov inequality, we have Then n −2 i∈I n s∈I n i =s Hence taking into account the convergence of the series of mixing coefficient, we have n −2 i∈I n s∈I n i =s hence it follows, using the assumption on the mixing coefficient, that The cases, i = s = j = u, i = u = s = j, j = s = i = u, t = u = i = s and s = u = i = j, are treated in the same manner as T 5 . 6-i = j = s = u |T 11 | = 1 n 2 i∈I n j∈I n s∈I n u∈I n Hence 7-i = j = s = u T 12 = n −2 i∈I n j∈I n i =j

It follows that
We obtain the same results for the following cases j = s = i = u, s = u = i = j, u = s = j = i. Taking into account all T i , we have Cov(I n,Z (ω), I n,Z ω ) = E(I n,Z (λ)I n,Z (λ )) − E I n,Z (λ)E I n,Z (λ ) In the following theorem, we establish the relation between the periodograms of {X t } and {Z t }.

Theorem 2 Under Assumptions
where Proof Let J X (ω) and J Z (ω) the discrete Fourier transform of {X t } and {Z t }, respectively. Then Note by and Then Note that we have two cases: In the first case, that is, | j k | < n k , we have, by Theorem 2.1 of [10] and under Assumptions 1 and 2 with ρ = 2, Since 0 < P < n, P → ∞ and under Assumption 2, we have P κ ∞ t=P t N −1 α(t) In the second case, there is n * terms, in the same manner as before, we have 1+δ) ).
In the first case, that is, | j k | < n k , by Assumptions 1 and 2 with ρ = 4 and applying Theorem 2.1 in [10] again, we have Since 0 < P < n, P → ∞ and under Assumption 2, we have P κ The second case is treated in the same manner, that is, there is n * terms, as in the first case we have Since n * ≤ 2 N | j|, then we find again the same result as in the first case.
Then in both cases, we have Since j∈Z N a j j As in [6] page 347, by applying Cauchy-Shwartz inequality and results on the covariance of the periodogram of innovations in Theorem 1, we obtain The following theorem provides the asymptotic properties of the periodogram of (X t ).
Since f X (g(n, λ)) → f X (λ) and R n (g(n, λ)) → 0 in probability, the result of the first part of the theorem is obtained from Theorem 1.
(ii) The proof is similar to the proof of Theorem 2.2 in [5], then omitted. It suffices to see that We have V ar(R n (λ k )) ≤ E(R n (λ k )) 2 and V ar(I n,Z (λ k )) is bounded uniformly in λ k . Using Cauchy-Schwartz inequality and since from Theorems 2 and 1, respectively, we have sup ω E |R n (λ)| 2 = O( n _1 ) and V ar(I n,Z (λ k )) = O( n _1 ), and we have
Then from Theorem 1, we have V ar(I n,X )(λ k ) if λ j = λ l and 0 < λ ji < π, ∀i = 1 to N and w ji s k = kπ .

Convergence of the smoothed periodogram
Since the periodogram estimator is not consistent, we introduce a linear filter and obtain a smoother periodogram. In this section, we study the convergence properties of this smoother periodogram estimator. Let (X t , t ∈ Z N ) be the linear process defined above, consider the following class of estimators such that ω j = ( 2π j 1 n 1 , . . . , 2π j N n N ) and where l n r → ∞, n r → ∞ and l n r < cn r for some c ∈ [0, 1]. (W n , n ∈ N N ) is a sequence of symmetric weight functions such that W n (k) are obtained as product of one-dimensional windows: Since l n r → ∞, n r → ∞ and l nr n r → 0, then max |k r |≤l nr ,r =1,...,N g(n, ω) + ω k − ω → 0, Since f is continuous, and for n large enough (see Proposition 10.3.1 in [6]), Since |k r |<l nr r =1,...N W n (k) = 1, we have

Numerical experiments
In this section, we propose some numerical results of the proposed periodogram towards some simulations. We consider a two-dimensional space (N = 2) with the process {X i, j , (i, j) ∈ Z 2 } simulated on a rectangular domain I = {(i, j), 1 ≤ i ≤ n 1 , 1 ≤ j ≤ n 2 } of n 1 × n 2 sites, we take without loss of generality n 1 = n 2 for different values of n 1 ∈ {30, 50, 100, 120}. The process is defined by .08 and f 3 = 0.05, U k i, j , k = 1, .., 3, and are independently uniformly distributed on [0, 1]. All of the following numerical analyses were carried out using the R software (version 3.5.1). Examples of simulated spatial process are given in Fig. 1 for n 1 = 100 and θ = 0.5, 0.9. This figure shows a higher spatial correlation for the process with higher parameter θ = 0.9. To investigate the finite sample properties of the periodogram and a smooth version, we take different values of the parameter θ ∈ {0.2, 0.5, 0.9} and two different sample sizes. Taking 300 spatial points s = (s 1 , s 2 ) ∈ [0, 1] 2 , we compute the periodogram and a smooth version at 300 frequencies at points 2sπ including f i , i = 1, . . . , 3. The results are given in Fig. 2. It clearly shows the higher frequency estimates of f 1 = 2, f 2 = 0.08 and f 3 = 0.05 defined in the model, particularly for large sample sizes, n 2 1 = 10 4 andn 2 1 = 144 * 10 2 . Note that in order to have the smooth spectral density, we use a weight function obtained by two successive applications of a Kolmogorov-Zurbenko filtering (package 'kzfs', see [14] for more details).
To assess the performance of the smooth periodogram estimation, particularly at the frequencies f i , i = 1, . . . , 3, we consider 100 replications of the simulated model and provide mean estimates. More precisely, we computef j = 1 100 100 q=1f q j , where thef q j are the high frequency estimates of f 1 = 2, f 2 = 0.08 and f 3 = 0.05, from the smooth periodogram computed with the q−th replicated sample. The obtained results are summarized in Table 1 that shows the results of Fig. 2, the good performance of the smooth periodogram estimation for large sample sizes.

Conclusion
In this paper, we have focused on the asymptotic properties of the periodogram of a linear spatial process with α mixing innovations. A relation between periodogram of innovations and the periodogram of the linear process is established in spatial case. As the estimator of the spectral density is inconsistent, a linear filter was introduced and the convergence of the smoothed estimate was studied. These studies have shown that the obtained results are similar to those obtained by [5,6] but in a spatial context.
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