Cross Spectral Analysis of a Gaussian Vector Process in the Presence of Variance Fluctuations

Abstract

Let $x'(t) = (x_1(t),x_2(t)),\quad(t = 1, 2, \cdots)$ be a two dimensional, Gaussian, vector process. Let the process $x'(t)$ have the representation \begin{equation*}\tag{1.1}x'(t) = \sum^p_{m = 0}B_my (t - m),\end{equation*} where \begin{equation*}\begin{align*}\tag{1.2} B_m &= \{b_{ijm}; i,j = 1, 2\}; \\ y'(t) &= (y_1(t), y_2(t)); \\ y_l(t) &= \sigma_l(t)\epsilon_l(t)\quad (l = 1, 2)\end{align*}.\end{equation*} The random variables $\epsilon_l(t)$ are independently and normally distributed with mean zero and variance unity. $p$ is a finite positive integer. The coefficients $B_m = (b_{ijm})_{2 \times 2}$ are finite real constants, and $\sigma^2_l(t)$ are non-random sequence of positive numbers which are not, in general equal, but do satisfy the conditions \begin{equation*}\tag{1.3}N^{-1}\sum^N_{t = 1}\sigma^2_l(t) = \nu_l < \infty\quad (\text{as} N \rightarrow \infty),\end{equation*} and $L \leqq \sigma^2_l(t) \leqq U < \infty\quad (t = 1, 2, \cdots).$ The relation (1.1) is a multivariate representation of a finite moving average process with time trending coefficients. Consider the matrix \begin{equation*}\begin{align*}\tag{1.4}F (\lambda) &= \begin{pmatrix}f_{11}(\lambda) & f_{12}(\lambda) \\ f_{21}(\lambda) & f_{22}(\lambda)\end{pmatrix} \\ &= G(\lambda)\begin{pmatrix}\nu_1 & 0 \\ 0 & \nu_2\end{pmatrix} G^{\ast'}(\lambda)\end{align*},\end{equation*} where $G(\lambda) = \sum^p_{m = 0} B_me^{im\lambda}$ and $G^\ast(\lambda)$ is its complex conjugate. Under the condition (1.3), Herbst [1] has defined $f_{11}(\lambda)$ and $f_{22}(\lambda)$ as the spectral densities of the processes $x_1(t)$ and $x_2(t)$ respectively, and considered their estimation. Here we generalize Herbst [1] results to a vector process and show that under the conditions (1.3) and (3.3) $f_{12}(\lambda)$, which is defined as the cross spectral density of the process $x_1(t)$ and $x_2(t)$, can consistently be estimated.