20. Gaussian Measures - Probability

Tutorial 20: Gaussian Measures120. Gaussian MeasuresMn(R)isthesetofalln n-matrices with real entries,n nition 141 AmatrixM2Mn(R)is said to besymmetric,if and only ifM= ,ifandonlyifMisnon-singular andM 1= symmetric, we say thatMisnon-negative, if and only if:8u2Rn;hu;Mui 0 Theorem 131 Let 2Mn(R),n 1, be a symmetric and non-negative real matrix. There exist 1;:::; n2R+andP2Mn(R)orthogonal matrix, such that: =P:0B@ n1CA:PtIn particular, there existsA2Mn(R)such that = 20: Gaussian Measures2As a rare exception, theorem (131) is given without 1andM2Mn(R), show that we have:8u;v2Rn;hu;Mvi=hMtu;viExercise 2Mn(R) be a symmetricand non-negative matrix. Let 1be the Probability measure onR:8B2B(R); 1(B)=1p2 ZBe x2=2dxLet = 1 ::: 1be the product measure (R)be such that =A:At. Wede nethemap :Rn!Rnby:8x2Rn; (x)4=Ax+m1. Show that is a Probability measure on (Rn;B(Rn)).2. Explain why the image measureP= ( ) is well-de Show thatPis a Probability measure on (Rn;B(Rn)).

Tutorial 20: Gaussian Measures 4 De nition 142 Let n 1 and m 2Rn.Let 2M n(R) be a symmetric and non-negative real matrix. The probability measure N n(m;) on Rnde ned in theorem (132) is called the n-dimensional gaussian measure or normal distribution,withmeanm2Rn and covariance matrix .

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  Distribution, Measure, Normal, Gaussian, Normal distribution, Gaussian measures

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