Positive Definite Kernels and Random Sequences Connected to Polynomial Hypergroups

TL;DR

This paper introduces new classes of nonstationary stochastic sequences linked to polynomial hypergroups, analyzing their covariance structures with positive definite kernels and developing efficient prediction algorithms and spectral analysis methods.

Contribution

It presents novel nonstationary sequence classes, new estimators for covariance, and a comprehensive prediction theory with efficient algorithms and spectral detection techniques.

Findings

New classes of nonstationary stochastic sequences identified

Efficient Levinson-type algorithms developed for prediction

Wiener-type theorems enable spectral measure atom detection

Abstract

This work explores new classes of nonstationary stochastic sequences associated with polynomial hypergroups. Their covariance structures are analyzed through positive definite kernels and corresponding Hilbert spaces. Novel consistent estimators are introduced for deriving covariance structures from sequence realizations. A comprehensive prediction theory is developed, including a fast Levinson-type algorithm for efficiently calculating best linear predictors. Wiener-type theorems are established, enabling the detection of spectral measure atoms via generalized periodograms. Additional advancements, such as prediction with supplementary information, further enhance the scope of this study.