random fields

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1、RandomFieldsILectureNotes(Draft)Prof.Dr.EvgenySpodarevUlm2009Contents1Basicnotionsofthetheoryofrandomfunctions11.1Randomfunctions...................................11.2Elementaryexamples..................................71.3Momentsandcovariance................................141.4Stationarityandiso

2、tropy................................171.5Continuityanddierentiability............................181.6ProofoftheTheoremofKolmogorov.........................241.7Additionalexercises..................................272Correlationtheoryofstationaryrandomelds322.1Positivesemi-denitefunctions......

3、......................322.1.1Isotropiccase..................................362.1.2Constructionprinciplesofpositivesemi-denitefunctions.........372.1.3Sucientconditionsforpositivesemi-deniteness..............382.1.4Examples....................................392.2Variograms......................

4、.................412.3Stochasticintegration.................................442.3.1Independentlyscatteredrandommeasures..................442.3.2Stochasticintegral...............................482.4Spectralrepresentationforstationaryrandomfunctions..............542.5Orthogonalexpansionsforrandomfun

5、ctions.....................592.5.1Mercer'sTheorem...............................592.5.2ReproducingKernelHilbertSpaces......................622.5.3Canonicalisomorphism............................662.5.4Karhunen-Loéveexpansion..........................672.6Additionalexercises.........................

6、.........733Modelsofrandomelds783.1Gaussianrandomelds.................................783.1.1PropertiesofpathsofGaussianrandomfunctions..............783.1.2Gaussianrandompolynomials.........................86Contents95Index100Index100i1Basicnotionsofthetheoryofrandomfunctions1.1RandomfunctionsLet

7、(Ω,F,P)beaprobabilityspace,Ω6=∅,and(S,B)beameasurablespaceconstructeduponanabstractsetS6=∅.Denition1.1.1ArandomelementX:Ω→SisanF

8、B-measurablemappingof(Ω,F)into(S,B),i.e.X−1(B):={ω∈Ω:X(ω)∈B}∈FforallB∈B.Wewr