## The Annals of Statistics

- Ann. Statist.
- Volume 1, Number 2 (1973), 322-333.

### Approximating Circulant Quadratic Forms in Jointly Stationary Gaussian Time Series

Leon Jay Gleser and Marcello Pagano

#### 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 $\{X(t)\}$.

#### Article information

**Source**

Ann. Statist., Volume 1, Number 2 (1973), 322-333.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342369

**Digital Object Identifier**

doi:10.1214/aos/1176342369

**Mathematical Reviews number (MathSciNet)**

MR362777

**Zentralblatt MATH identifier**

0256.62083

**JSTOR**

links.jstor.org

**Keywords**

6285 6050 Periodogram asymptotic distributions complex Wishart

#### Citation

Gleser, Leon Jay; Pagano, Marcello. Approximating Circulant Quadratic Forms in Jointly Stationary Gaussian Time Series. Ann. Statist. 1 (1973), no. 2, 322--333. doi:10.1214/aos/1176342369. https://projecteuclid.org/euclid.aos/1176342369