Open Access
2012 Szegö’s theorem and its probabilistic descendants
N.H. Bingham
Probab. Surveys 9: 287-324 (2012). DOI: 10.1214/11-PS178


The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szegö’s work of 1915-21, and has been given a great impetus by the recent work of Simon, in particular his survey paper and three recent books; we allude to the title of the third of these, Szegö’s theorem and its descendants, 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 classic book by Grenander and Szegö, Toeplitz forms and their applications. Coming to the subject from this background, our aim here is to complement this recent work by giving some probabilistically motivated results. We also advocate a new definition of long-range dependence.


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N.H. Bingham. "Szegö’s theorem and its probabilistic descendants." Probab. Surveys 9 287 - 324, 2012.


Published: 2012
First available in Project Euclid: 23 July 2012

zbMATH: 1285.60037
MathSciNet: MR2956573
Digital Object Identifier: 10.1214/11-PS178

Primary: 60G10
Secondary: 60G25

Keywords: autoregressive , cepstrum , Hardy space , long-range dependence , moving average , orthogonal polynomials on the unit circle , partial autocorrelation function , Prediction theory , stationary process

Rights: Copyright © 2012 The Institute of Mathematical Statistics and the Bernoulli Society

Vol.9 • 2012
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