High Dimensional Probability III

ISBN-13: 9783034894234 / Angielski / Miękka / 2012 / 346 str.

Joergen Hoffmann-Joergensen; Michael B. Marcus; Jon A. Wellner

Opis

The title High Dimensional Probability is used to describe the many tributaries of research on Gaussian processes and probability in Banach spaces that started in the early 1970s. Many of the problems that motivated researchers at that time were solved. But the powerful new tools created for their solution turned out to be applicable to other important areas of probability. They led to significant advances in the study of empirical processes and other topics in theoretical statistics and to a new approach to the study of aspects of Levy processes and Markov processes in general. The papers in this book reflect these broad categories. The volume thus will be a valuable resource for postgraduates and reseachers in probability theory and mathematical statistics.

Spis treści

I. Measures on General Spaces and Inequalities.- Stochastic inequalities and perfect independence.- Prokhorov-LeCam-Varadarajan’s compactness criteria for vector measures on metric spaces.- On measures in locally convex spaces.- II. Gaussian Processes.- Karhunen-Loeve expansions for weighted Wiener processes and Brownian bridges via Bessel functions.- Extension du thèoréme de Cameron-Martin aux translations aleatoires. II. Intègrabilitè des densitès.- III. Limit Theorems.- Rates of convergence for Lèvy’s modulus of continuity and Hinchin’s law of the iterated logarithm.- On the limit set in the law of the iterated logarithm for U-statistics of order two.- Perturbation approach applied to the asymptotic study of random operators.- A uniform functional law of the logarithm for a local Gaussian process.- Strong limit theorems for mixing random variables with values in Hilbert space and their applications.- IV. Local Times.- Local time-space calculus and extensions of Ito’s formula.- Local times on curves and surfaces.- V. Large, Small Deviations.- Large deviations of empirical processes.- Small deviation estimates for some additive processes.- VI. Density Estimation.- Convergence in distribution of self-normalized sup-norms of kernel density estimators.- Estimates of the rate of approximation in the CLT for L1-norm of density estimators.- VII. Statistics via Empirical Process Theory.- Statistical nearly universal Glivenko-Cantelli classes.- Smoothed empirical processes and the bootstrap.- A note on the asymptotic distribution of Berk-Jones type statistics under the null hypothesis.- A note on the smoothed bootstrap.