LEADER 03877nam 22005415 450 001 996466611903316 005 20200707022746.0 010 $a3-642-37617-7 024 7 $a10.1007/978-3-642-37617-7 035 $a(CKB)3710000000002579 035 $a(SSID)ssj0000962999 035 $a(PQKBManifestationID)11525797 035 $a(PQKBTitleCode)TC0000962999 035 $a(PQKBWorkID)10979212 035 $a(PQKB)11143314 035 $a(DE-He213)978-3-642-37617-7 035 $a(MiAaPQ)EBC3107040 035 $a(PPN)172426421 035 $a(EXLCZ)993710000000002579 100 $a20130704d2013 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt 182 $cc 183 $acr 200 10$aOn the Estimation of Multiple Random Integrals and U-Statistics$b[electronic resource] /$fby Péter Major 205 $a1st ed. 2013. 210 1$aBerlin, Heidelberg :$cSpringer Berlin Heidelberg :$cImprint: Springer,$d2013. 215 $a1 online resource (XIII, 288 p. 11 illus.) 225 1 $aLecture Notes in Mathematics,$x0075-8434 ;$v2079 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a3-642-37616-9 327 $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? ?ervonenkis 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. 330 $aThis work starts with the study of those limit theorems in probability theory for which classical methods do not work. In many cases some form of linearization can help to solve the problem, because the linearized version is simpler. But in order to apply such a method we have to show that the linearization causes a negligible error. The estimation of this error leads to some important large deviation type problems, and the main subject of this work is their investigation. We provide sharp estimates of the tail distribution of multiple integrals with respect to a normalized empirical measure and so-called degenerate U-statistics and also of the supremum of appropriate classes of such quantities. The proofs apply a number of useful techniques of modern probability that enable us to investigate the non-linear functionals of independent random variables. This lecture note yields insights into these methods, and may also be useful for those who only want some new tools to help them prove limit theorems when standard methods are not a viable option. 410 0$aLecture Notes in Mathematics,$x0075-8434 ;$v2079 606 $aProbabilities 606 $aProbability Theory and Stochastic Processes$3https://scigraph.springernature.com/ontologies/product-market-codes/M27004 615 0$aProbabilities. 615 14$aProbability Theory and Stochastic Processes. 676 $a519.24 700 $aMajor$b Péter$4aut$4http://id.loc.gov/vocabulary/relators/aut$048603 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466611903316 996 $aOn the estimation of multiple random integrals and U-statistics$9258678 997 $aUNISA