Autocorrelation function bias owed to a limited number of de-trended observations. Applications to autoregressive models with noise.
Analytical expressions were derived for the expectation of the autocorrelation function (ACF) corresponding to low-frequency least squares fits and deviations from them in the case of a limited number of observations N. A vector of values of the autocorrelation function may be obtained by multiplication of a N N matrix Z (dependent on concrete basic functions used for trend determination) by a vector of the unbiased ACF. Because much computational time is needed to obtain such a matrix, its components are to be computed once for concrete N and basic functions, and then stored as a file. An algorithm is proposed for determining the contribution of the correlated signal to the 'signal noise'. The expressions are written for general form of the ACF, and illustrated by the application to autoregressive models. The statistical properties of the model parameters are studied. The method is applied to cataclysmic binaries AM Her and TT Ari. The precise expressions allow us to obtain reliable results and to avoid misinterpretation being possible when using the approximate methods.