# On the Estimation of Multiple Random Integrals and U-Statistics

Part of the Lecture Notes in Mathematics book series (LNM, volume 2079)

Part of the Lecture Notes in Mathematics book series (LNM, volume 2079)

This work starts with the study of those limit theorems in probability theory for which classical methods do not work. In many cases some form of linearization can help to solve the problem, because the linearized version is simpler. But in order to apply such a method we have to show that the linearization causes a negligible error. The estimation of this error leads to some important large deviation type problems, and the main subject of this work is their investigation. We provide sharp estimates of the tail distribution of multiple integrals with respect to a normalized empirical measure and so-called degenerate *U*-statistics and also of the supremum of appropriate classes of such quantities. The proofs apply a number of useful techniques of modern probability that enable us to investigate the non-linear functionals of independent random variables.

This lecture note yields insights into these methods, and may also be useful for those who only want some new tools to help them prove limit theorems when standard methods are not a viable option.

This lecture note yields insights into these methods, and may also be useful for those who only want some new tools to help them prove limit theorems when standard methods are not a viable option.

60H05, 60G60, 60F10 Chaining argument Large deviation estimates Vapnik–Červonenkis classes Wiener–Itô integrals and diagram formula

- DOI https://doi.org/10.1007/978-3-642-37617-7
- Copyright Information Springer-Verlag Berlin Heidelberg 2013
- Publisher Name Springer, Berlin, Heidelberg
- eBook Packages Mathematics and Statistics
- Print ISBN 978-3-642-37616-0
- Online ISBN 978-3-642-37617-7
- Series Print ISSN 0075-8434
- Series Online ISSN 1617-9692
- About this book