A note on maximum entropy. (English) Zbl 0682.60030

Rosenblatt, Murray

Probability, statistics, and mathematics, Pap. in Honor of Samuel Karlin, 255-260 (1989).

[For the entire collection see Zbl 0679.00013.]
\(X_1, \ldots, X_{m-1}\) is a stationary sequence of random variables with covariance sequence \(c_ t = \cos w_ m t + \varepsilon \delta_{0,t}\), \(t=0,\ldots,m-1\) where \(w_ m = 2\pi k/m\), \(\varepsilon>0\), \(k\) is an integer between 0 and \(m-1\) inclusive, and \(\delta\) is the usual Kronecker symbol. Define \(a_0, a_1, \ldots, a_{m-1}\) by the matrix equation \[ \left[ \begin{matrix} c_0 & c_1 & c_2 & \dots & c_{m-1} \\ c_1 & c_0 & c_1 & \dots & c_{m-2} \\ c_2 & c_1 & c_0 & \dots & c_{m-3} \\ \vdots & \vdots & \vdots && \vdots \\ c_{m-1} & c_{m-2} & c_{m-3} & \dots & c_0 \end{matrix} \right] \left[ \begin{matrix} 1 \\ a_1 \\ a_2 \\ \vdots \\ a_{m-1} \end{matrix} \right] = \left[ \begin{matrix} a_0 \\ 0 \\ 0 \\ \vdots \\ 0 \end{matrix} \right] \] and the power spectrum \(P(z) = a_0 \left| 1 + \sum^{m-1}_{j=1} a_ j z^ j \right|^{-2}\). It is shown that if \(m\) is large, a first-order approximation to \(P(\exp(-2\pi i s/m))\) is \(m/4\varepsilon\) if \(s=k\), \(m-k\), and is \(\varepsilon\) if \(s\) is any other integer between 0 and \(m-1\) inclusive.

Reviewer: L.Weiss

MSC:

60G35	Signal detection and filtering (aspects of stochastic processes)
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15	Inference from stochastic processes and spectral analysis

Keywords:

maximum entropy; power spectrum; stationary sequence of random variables

Citations:

Zbl 0679.00013