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020 _a9783642376177
_9978-3-642-37617-7
024 7 _a10.1007/978-3-642-37617-7
_2doi
050 4 _aQA273.A1-274.9
050 4 _aQA274-274.9
072 7 _aPBT
_2bicssc
072 7 _aPBWL
_2bicssc
072 7 _aMAT029000
_2bisacsh
082 0 4 _a519.2
_223
100 1 _aMajor, Péter.
_eauthor.
245 1 0 _aOn the Estimation of Multiple Random Integrals and U-Statistics
_h[electronic resource] /
_cby Péter Major.
264 1 _aBerlin, Heidelberg :
_bSpringer Berlin Heidelberg :
_bImprint: Springer,
_c2013.
300 _aXIII, 288 p. 11 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2079
505 0 _a1 Introduction -- 2 Motivation of the investigation. Discussion of some problems -- 3 Some estimates about sums of independent random variables -- 4 On the supremum of a nice class of partial sums -- 5 Vapnik– Červonenkis classes and L2-dense classes of functions -- 6 The proof of Theorems 4.1 and 4.2 on the supremum of random sums -- 7 The completion of the proof of Theorem 4.1 -- 8 Formulation of the main results of this work -- 9 Some results about U-statistics -- 10 MultipleWiener–Itô integrals and their properties -- 11 The diagram formula for products of degenerate U-statistics -- 12 The proof of the diagram formula for U-statistics -- 13 The proof of Theorems 8.3, 8.5 and Example 8.7 -- 14 Reduction of the main result in this work -- 15 The strategy of the proof for the main result of this work -- 16 A symmetrization argument -- 17 The proof of the main result -- 18 An overview of the results and a discussion of the literature.
520 _aThis 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.
650 0 _aMathematics.
650 0 _aDistribution (Probability theory).
650 1 4 _aMathematics.
650 2 4 _aProbability Theory and Stochastic Processes.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783642376160
830 0 _aLecture Notes in Mathematics,
_x0075-8434 ;
_v2079
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-642-37617-7
912 _aZDB-2-SMA
912 _aZDB-2-LNM
999 _c98128
_d98128