N. H. Bingham

Fluctuation theory in continuous time

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 Levy 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.

Continuous branching processes and spectral positivity

Certain properties of continuous-state branching processes are studied via the random time-change linking them with spectrally positive Levy processes. The results are compared and contrasted with those for simple branching processes.

Szegö's theorem and its probabilistic descendants

The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szegos 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, Szegos theorem and its descendants , in ours. Simons 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 Szego, 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.

Gaussian processes on compact symmetric spaces

Paul Lévy studied Gaussian processes ξ(a) with the parameter a running over Euclidean d-space Rd and he also studied the case when a runs over the d-sphere Sd. His results were extended by Gangolli in a number of directions, one being the extension to the cases where the parameter a lies in the other two-point homogeneous Riemannian manifolds. In the compact cases Gangolli showed there was a distinction between spheres and projective spaces, in that the process discovered by Lévy which he called Brownian motion parametrized by spheres does not exist for projective spaces. However many interesting Gaussian process exist with parameters running through projective spaces as we show.

Modelling asset returns with hyperbolic distributions

Publisher Summary This chapter discusses applications of the hyperbolic distributions in financial modeling. To model the everyday movement of ordinary quoted stocks under the market pressure of many agents, an infinite measure is appropriate. The mixture representation transfers to characteristic functions on taking the Fourier transform. Hyperbolic densities provide a good fit for a range of financial data, not only in the tails but throughout the distribution. The hyperbolic tails are log-linear: much fatter than normal tails but much thinner than stable ones. Under the assumptions of independence and identical distribution, a maximum likelihood analysis is performed. There is more mass around the origin and in the tails than the normal distribution suggests and that fitting returns to a hyperbolic distribution is to be preferred. By contrast, the hyperbolic approach is designed to give a reasonable fit throughout and in particular a better fit overall than the normal.

Kingman, category and combinatorics

Focussing on the work of Sir John Kingman, one of the worlds leading researchers in probability and mathematical genetics, this book touches on the important areas of these subjects in the last 50 years. Leading authorities give a unique insight into a wide range of currently topical problems. Papers in probability concentrate on combinatorial and structural aspects, in particular exchangeability and regeneration. The Kingman coalescent links probability with mathematical genetics and is fundamental to the study of the latter. This has implications across the whole of genomic modelling including the Human Genome Project. Other papers in mathematical population genetics range from statistical aspects including heterogeneous clustering, to the assessment of molecular variability in cancer genomes. Further papers in statistics are concerned with empirical deconvolution, perfect simulation, and wavelets. This book will be warmly received by established experts as well as their students and others interested in the content.

On Higher-dimensional analogues of the ARC-sine law

The arc-sine laws form one of the cornerstones of classical one-dimensional fluctuation theory. In higher dimensions, knowledge of fluctuation theory remains a great deal less complete. Motivated by this, we consider higher-dimensional analogues of the classical arc-sine laws.

Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowskis theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

An explicit representation of Verblunsky coefficients

We prove a representation of the partial autocorrelation function (PACF) of a stationary process, or of the Verblunsky coefficients of its normalized spectral measure, in terms of the Fourier coefficients of the phase function. It is not of fractional form, whence simpler than the existing one obtained by the second author. We apply it to show a general estimate on the Verblunsky coefficients for short-memory processes as well as the precise asymptotic behavior, with remainder term, of those for FARIMA processes.

Multivariate prediction and matrix Szegö theory

Following the recent survey by the same author of Szegos theorem and orthogonal polynomials on the unit circle (OPUC) in the scalar case, we survey the corresponding multivariate prediction theory and matrix OPUC (MOPUC).