Approximating Circulant Quadratic Forms in Jointly Stationary Gaussian Time Series

Abstract

Let $\{X(t), t = 0, \pm 1, \pm 2,\cdots\}$ be a $p$-dimensional, zero mean, stationary Gaussian time series with matrix-valued covariance function $R(u) = (r_{jk}(u)), u = 0, \pm 1, \pm 2, \cdots; j, k = 1, 2,\cdots, p$. Let $F(\omega)$ be the spectral density matrix of the time series (if $F(\omega)$ exists), and assume that $F(\omega)$ is positive definite for all $\omega\in(0, 2\pi\rbrack$. Let $\hat{F}_X(\omega_l)$ be an estimator of $F(\omega_l)$ formed by averaging $(2n + 1)$ periodogram ordinates centered and equally spaced around $\omega_l, l = 1, 2,\cdots, M$, where the $\omega_l$'s themselves are equally spaced on $(0, \pi\rbrack$, and where all periodogram ordinates are based on the same record of length $T, (2n + 1)M \leqq T/2$, taken from the time series $\{X(t)\}$. Wahba (1968) has shown that if $\sum_j\sum_k\sum_u|ur_{jk}(u)| < \infty$ and $\log_2 M \leqq n$, then it is possible to construct $M$ independent complex Wishart matrices $W_T(\omega_l), l = 1, 2,\cdots, M$, such that $\{\hat{F}_X(\omega_l), l = 1, 2,\cdots, M\}$ converge simultaneously in mean square to $\{W_T(\omega_l), l = 1, 2,\cdots, M\}$ as $n, M$ (and thus $T$) get large. In the present paper, it is shown that Wahba's result holds under the less restrictive condition that $\sum_j\sum_k\sum_u|u|^{\frac{1}{2}}|r_{jk}(u)| < \infty$, and without our needing to assume that $\log_2 M \leqq n$. In consequence, a form of weak convergence of the averaged periodogram to a certain matrix-valued Wishart stochastic process is demonstrated (something, by the way, that Wahba (1968) cannot show because of her restriction that $\log_2 M \leqq n$). This result is a consequence of some general conclusions concerning the approximation of circulant quadratic forms in the time series $\{X(t)\}$.