Оцiнки кореляцiйної функцiї гауссового стацiонарного процесу, коли вiдомi його значення у скiнченнiй множинi точок

Abstract

In practice, process values are usually observed at certain points in time. And based on this data, we need to draw conclusions about the behavior of the process that is being monitored. That is why the primary purpose of the paper is to evaluate the covariance function of such a process. For this purpose, in this paper, we consider a Gaussian stationary random process $X$ with unknown mean, when its values are known in a finite set of points and the task is to estimate the covariance function of such a random process. One feature of estimating the correlation function of a random process with an unknown mean is that the use of correlograms as an estimator is not possible, since the correlogram in this case is a biased estimation of the correlation function. Therefore, to prove the theorems, it was necessary to construct a statistics that would be an unbiased estimate of the covariance function of a Gaussian stationary random process. In addition, as shown in some of our previous papers and in this work, we are dealing with quadratic-Gaussian processes when estimating the deviations of the correlation function of a Gaussian stationary random process from a correlogram in the $L_p$-metric. Therefore, to prove this estimate was used the theory of quadratic-Gaussian random processes. Using this theory, we obtain estimates of the deviations of the correlation function of a Gaussian stationary random process with an unknown mean, when its values in the finite set of points of this process from its estimate in $L_p$-metric are known. The paper also builds a criterion for testing the hypothesis of the appearance of the correlation function of such a random process. This criterion was formulated using the obtained estimates.