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GAUSSIAN RANDOM VECTORS AND PROCESSES

, , ,wellknown, ,theseassumptionsareoftenapproximatelysa tisfied,sotheresults,ifusedwithinsightan dcare, ,butstartswithastudyofGaussian(normal1)r andomvariablesandvectors, (rv)WisdefinedtobeanormalizedGaussianrvi fithasthedensityfW(w)=1p2 exp w22 ;forallw2R.( )1 Gaussianrv sareoftencallednormalrv ,firstbecausethecorrespondingprocessesar eusuallycalledGaussian,secondbecauseGaus sianrv s(whichhavearbitrarymeansandvariances)ar eoftennormalizedtozeromeanandunitvarianc e,andthird,becausecallingthemnormalgives thefalseimpressionthatotherrv (w)integratesto1( ,itisaprobabilitydensity), , ,ifweconsiderthervZ= W,thenthedistributionfunctionsofZandWare relatedbyFZ( w)=FW(w).Thismeansthattheprobabilitydens itiesarerelatedby fZ( w)=fW(w).ThusthePDFofZisgivenbyfZ(z)=1 fW z =1p2 exp z22 2 .( )ThusthePDFforZisscaledhorizontallybythe factor ,andthenscaledverticallyby1/ ( ).

This chapter is aimed primarily at Gaussian processes, but starts with a study of Gaussian (normal1) random variables and vectors, These initial topics are both important in their own right and also essential to an understanding of Gaussian processes. The material here is essentially independent of that on Poisson processes in Chapter 2.

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