Szeg\"o's Theorem and its Probabilistic Descendants

Abstract

The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szeg\"o's work of 1915-21, and has been given a great impetus by the recent work of Simon, in particular his two-volume book [Si4], [Si5], the survey paper (or summary of the book) [Si3], and the book [Si9], whose title we allude to in ours. Simon's motivation comes from spectral theory and analysis. Another major area of application of OPUC comes from probability, statistics, time series and prediction theory; see for instance the book by Grenander and Szeg\"o [GrSz]. Coming to the subject from this background, our aim here is to complement [Si3] by giving some probabilistically motivated results. We also advocate a new definition of long-range dependence.