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1、iNormalApproximationswithMalliavinCalculusFromStein’sMethodtoUniversalitybyIvanNourdinandGiovanniPeccatiiiToLili,JulietteandDelphine.ToEmmaEl¯ızaandIeva.iiiPrefaceThisisatextaboutprobabilisticapproximations,thataremathematicalstatementsprovidingestimatesonthedistancebetweenthelawsoftw
2、orandomobjects.Asthetitlesuggests,wewillbemainlyinterestedinapproximationsinvolvingoneormorenormal(equivalentlycalledGaussian)randomelements.NormalapproximationsarenaturallyconnectedwithCentralLimitTheorems(CLTs),i.e.convergenceresultsdisplayingaGaussianlimit,andareoneoftheleadingthem
3、esofthewholetheoryofProbability.Themainthreadofourtextconcernsthenormalapproximations,aswellasthecorre-spondingCLTs,associatedwithrandomvariablesthatarefunctionalsofagivenGaussianfield,likeforinstancea(fractional)Brownianmotionontherealline.Inparticular,apivotalrolewillbeplayedbytheele
4、mentsoftheso-calledGaussianWienerchaos.TheconceptofWienerchaosgeneralizestoaninfinite-dimensionalsettingthepropertiesoftheHermitepolynomials(thataretheorthogonalpolynomialsassociatedwiththeone-dimensionalGaus-siandistribution),andisnowacrucialobjectinseveralbranchesoftheoreticalandappl
5、iedGaussiananalysis.Thecornerstoneofourbookisthecombinationoftwoprobabilistictechniques,namelytheMalliavincalculusofvariationsandtheStein’smethodforprobabilisticapproximations.TheMalliavincalculusofvariationsisaninfinite-dimensionaldifferentialcalculus,whoseoperatorsactonfunctionalsofge
6、neralGaussianprocesses.InitiatedbyPaulMalliavin(start-ingfromtheseminalpaper[70],whichfocusedonaprobabilisticproofofH¨ormander“sumofsquares”theorem),thistheoryisbasedonapowerfuluseofinfinite-dimensionalintegrationbypartsformulae.Althoughoriginallyexploitedforstudyingtheregularityofthel
7、awsofWienerfunctionals(suchasthesolutionsofstochasticdifferentialequations),thescopeofitsactualapplications,rangingfromdensityestimatestoconcentrationinequalities,andfromanticipativestochasticcalculustothecomputationsof“Greeks”inmathematicalfinance,continuestogrow.Foraclassicpresentatio
8、nofth
