Zhurakovskyi B. The detection of hidden periodicities in regression models with locally transformed Gaussian stationary noise.

equencies joint least squares estimators in Walker sense of a sum of harmonics, joint periodogram estimators of harmonic oscillation parameters, joint periodogram estimators of amplitude and scale parameter of almost-periodic regression function are proved. The proof of the least squares estimator in Walker sense of nonlinear regression model vector parameter asymptotic normality consists of four steps. In the first step the reduction (linearization) theorem is proved. This theorem allows to replace the study of nonlinear regression parameter least squares estimator by the study of the correspondent estimator of some auxiliary linear regression model In the second step the theorem on asymptotic uniqueness in probability of weakly consistent least squares estimator. This fact is needed for correct application of Brouwer fixed point theorem on the last stage of the least squares estimator asymptotic normality proof. The third step is to obtain asymptotic normality of vector integral of nonlinearly locally transformed Gaussian stationary process weighted by regression function gradient. This central limit theorem in fact is the proof of the least squares estimator asymptotic normality of auxiliary linear regression model parameter that appears in the first step. In the proof of the central limit theorem the classic notion of regression function spectral measure is used. In the last step all the previous theorems are used to prove least squares estimator asymptotic normality. The fundamental moments in obtaining of this result are the the use of Brouwer fixed point theorem and von Bahr, Bhattacharya and Ranga Rao theorem on the measures of the layers of convex Borel sets in Rq for some class of measures that includes Gaussian measures.