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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).

A random variable U with this density, for arbitraryµ and 0, is defined to be a Gaussian random variable and is denoted U ⇠ N(µ,2). The added generality of a mean often obscures formulas; we usually assume zero-mean rv’s and random vectors (rv’s) and add means later if necessary. Recall that any rv U with a

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