We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the algebraic point of view of our recent work on Popa groups.

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Our aim here is to give a survey of that part of continuous-time fluctuation theory which can be approached in terms of functionals of Lévy processes, our principal tools being Wiener-Hopf factorisation and local-time theory. Particular emphasis is given to one- and two-sided exit problems for spectrally negative and spectrally positive processes, and their applications to queues and dams. In addition, we give some weak-convergence theorems of heavy-traffic type, and some tail-estimates involving regular variation.

We obtain results connecting the distributions of the random variables Z 1 and W in the supercritical Galton-Watson process. For example, if a > 1, and converge or diverge together, and regular variation of the tail of one of Z 1, W with non-integer exponent α > 1 is equivalent to regular variation of the tail of the other.

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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 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 Szegö [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.AMS 2000 subject classifications. Primary 60G10, secondary 60G25.