Stationary Random Fields on the Unitary Dual of a Comoact Group

TL;DR

This paper extends the concept of stationarity from sequences to random fields indexed by the unitary dual of compact groups, exploring spectral measures, white noise, and connections to hypergroup theory.

Contribution

It generalizes stationarity to fields indexed by the unitary dual of compact groups, including spectral analysis and construction of stationary time series.

Findings

Covariance functions are positive definite and Fourier transform of finite central measures.

Extended classical theorems like Cramer and Kolmogorov to this new framework.

Constructed examples of stationary fields and white noise on certain groups.

Abstract

We generalise the notion of wide-sense stationarity from sequences of complex-valued random variables indexed by the integers, to fields of random variables that are labelled by elements of the unitary dual of a compact group. The covariance is positive definite, and so it is the Fourier transform of a finite central measure (the spectral measure of the field) on the group. Analogues of the Cramer and Kolmogorov theorems are extended to this framework. White noise makes sense in this context and so, for some classes of group, we can construct time series and investigate their stationarity. Finally we indicate how these ideas fit into the general theory of stationary random fields on hypergroups.