Abstract

The explicit representation for the limiting spectral moments of sample covariance matrices generated by the periodic autoregressive model (PAR) is established. We propose to use the moment-constrained maximum entropy method to estimate the spectral density function. The experiments show that the maximum entropy spectral density function curve obtained based on the fourth-order limiting spectral moment can match histograms of the eigenvalues of the covariance matrices very well.

1. Introduction

Periodic time-series models are used for modeling stationary time series with periodic characteristics. Periodicity refers to wave or oscillating movements around long-term trends presented in the time series, such as the water sector, where the supply of water varies with seasonal rainfall. In the economic time series, it is often accompanied by some cycle or quasi-periodic fluctuations, such as day and night changes of urban electricity consumption, seasonal changes of railway passenger transport, and repeated acceleration, interruption, stagnation, and recovery of economic growth; see Franses and Paap [1], Bell et al. [2], and Aliat and Hamdi [3]. The periodic autoregressive (PAR) model is a generalisation of the classical autoregressive (AR) model by allowing the parameters to vary with the variables over a period:

For a given and a predetermined value , the random variable denotes the value in the quarter of the cycle, where and . In the cycle, is an AR process. The model order is given by , and , represents the AR model coefficients for the quarter. The error process corresponds to a periodic white noise, with and . Let , where the transpose (indicated by ) of a row vector is a column vector, and we define the large-dimensional sample covariance matrices as , with the periodic length tends to infinity in proportion to the sample size , namely, . Sample covariance matrices are very important in multivariate statistical inference since many test statistics are defined by their eigenvalues or functions.

Random matrix theory (RMT) originated from the study of the energy levels of a large number of particles in quantum mechanics. Many of the laws of mathematical physics were discovered through numerical studies. In the late 1950s, E. P. Wigner formulated the problem in terms of the empirical distribution of random matrices [4, 5], which led to the study of the semicircular law of Gaussian matrices. Since then, RMT has formed an active branch in modern probability theory. The LSD of the sample covariance matrix of a large-dimensional random matrix composed of independent random variables, the MP law, was first established by Marcenko and Pastur [6]. Further research efforts were conducted to estimate the LSD of a product of two random matrices. Yin and Krishnaiah [7] proved the existence of LSD for the matrix sequence when the two matrix sequences and are standard Wishart matrices and positive definite matrices, respectively. In particular, Bai et al. [8] obtained a display representation of the LSD of the matrix sequence when the two matrix sequences and are the sample covariance matrix and the Wigner matrix, respectively. To relax the independence of the entries of , Silverstein [9] considered the case of , where is a nonnegative definite matrix and consists of independent and identically distribution (i.i.d.) entries, and the sample covariance matrices take the form of . Random matrices of the form were extensively investigated by many researchers, including Yin [10], Silverstein [9], Bai and Zhou [11], Jin et al. [12], Zhang et al. [13] and Yao [14]. Silverstein’s important work [9] aimed to relax the independence structure between coordinates and considered random vectors of the form , assuming that the spectral norm of the matrix sequence is bounded and that the ESD of the matrix sequence is convergent. Silverstein proved that the LSD of the sample covariance matrix exists, given by the LSD of equation representation of the Stieltjes transformation.

This paper investigates the LSD of large-dimensional sample covariance matrices generated by the PAR model, and the sample covariance matrices take the form of . A similar ARMA-type processes problem has been considered by Bai and Zhou [11], Jin et al. [12], Yao [14], etc. It follows that, in the ARMA-type processes, the covariance matrices equals to the Toeplitz matrix, according to the fundamental eigenvalue distribution theorem for Toeplitz-type matrices [15], the ESD of weakly converges to the nonrandom distribution H. For some situations, the ESD of have explicit form; then, the LSD for the sample covariance matrix can be solved explicitly. For the random matrix generated by the first-order vector autoregressive (VAR(1)) model and the first-order vector moving average (VMA(1)) model, Jin et al. [12] and Bai and Zhou [11] obtained a display representation of this LSD. However, in our first-order variable-coefficient PAR process, we consider the process be interrupted by certain factors that the odd terms and the even terms of the coefficient converge to and , respectively, and that . The mixing matrix is a lower triangular matrix whose main diagonal elements are all 1 and subdiagonal elements differ from each other. When , the limiting form of the matrices are Toeplitz matrices, and the eigenvalues of Toeplitz matrices have been proved by Gregory [16]; when , the ESD of has no explicit form, see [17, 18], and the structure of the limiting matrices are very complex. Our main contributions in this paper include the following. (i) For the first-order variable-coefficient PAR model, the explicit forms of limiting spectral moments for are given; through verifying the Carleman condition, we derive the explicit spectral moments of . (ii) A framework of the maximum entropy-based method is provided for estimate the LSD.

The paper is organized as follows. In Section 2, we employ the moment method to deduce the spectral density of sample covariance matrices. In Section 3, we use the maximum entropy method to find the approximate solution and analyze the convergence of the maximum entropy solution. We simulate the PAR model to obtain the periodic sample covariance matrices, and the experiment shows that the maximum entropy spectral density function curve can match histograms of the eigenvalues of the covariance matrices very well.

2. Methodology and Main Results

In this section, we first derive the explicit spectral moments form of ; subsequently, we verify the spectral moments statistics satisfying the Carleman condition, and then, we employ the moment method to deduce the spectral density of sample covariance.

We consider first-order PAR model:

Let

Then, the first-order PAR model can be defined aswhere denotes the equation on the left can push the equation on the right. From equation (3), is a lower triangular matrix whose main diagonal elements are all 1, and therefore, the matrix is invertible. Hence,

The sample covariance matrices and have the same ESD. Let , and the ESD of is the same as that of . Next, we establish the existence of the LSD of the covariance matrix sequence .

Theorem 1. Let be generated by a first-order PAR model in equation (2), and the odd terms and the even terms of the coefficient converge to and , respectively, and . The error is independent random variables satisfying for any , ; furthermore, we assume that ; then, the spectral moments of sample covariance matrix tend towhere is the limit of spectral moments of , which have an explicit form:

Proof. The proof is divided into two steps. We first show the existence of the limit of spectral moments which is defined as ; then, we verify the Carleman condition. From equation (3), the inverse of matrix can be written aswhere can be written asLetThen,Furthermore, we obtain thatApplying the trace operator, we haveBased on the assumption that , we obtainand therefore,LetThen,The following section wishes to obtain a simplified form of the function . From equations (10) and (16), we haveDefineThen, can be rewritten asLetTherefore, can have a simplified representation asBy the derivation operations,we obtain thatBy partially replacing in equation (24) with , we define a new functionDefineby whichsoThen,We use to take place of and obtain thatTherefore,Next, we verify the Carleman condition .
From equation (30), it follows thatThen, applying the equality and , we obtainFrom Stirling’s formula , we get . Then, ; applying this inequality, has a finite lower bound:Then,With the moment convergence theorem, the spectral distribution of tends to a nonrandom probability distribution H. Let and ; by the important Theorem 2.10 of Bai [19], the ESD of tends to a nonrandom limit distribution in probability (or almost surely). The spectral moments of satisfy the following equation of Yin [10]:From the moment convergence theorem, we can obtain that the ESD of converges with probability close to 1 to a nonrandom distribution, determined by the spectral moment statistic.