Introduction

Abstract

Traditionally the most important problem of mathematical statistics dealing with random stationary processes X_t, t = …,-1,0,1, … is the problem of estimating the second order characteristics, namely the covariance function

$$\beta \left( T \right) = E\{ [{X_t} - E({X_t})][{X_{t + T}} - E\left( {{X_{t + T}}} \right)]\} $$

or its Fourier transform — the spectral density f = f(λ) (under the assumption that the spectral density exists). For this reason, a vast amount of periodical and monographic literature is devoted to the nonparametric statistical problem of estimating the function β(τ) and especially that of f(λ) (see, for example, the books [4,21,22,26,56,77,137,139,140,]). However, the empirical value f ^*_n of the spectral density f obtained by applying a certain statistical procedure to the observed values of the variables X₁, … , X_n, usually depends in a complicated manner on the cyclic frequency λ. This fact often presents difficulties in applying the obtained estimate f ^*_n of the function f to the solution of specific problems related to the process X_t.