## A note on maximum entropy.(English)Zbl 0682.60030

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

Zbl 0679.00013