20. Gaussian Measures - Probability

1 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)).

2 20: Gaussian Measures34. Show that for allu2Rn:FP(u)=ZRneihu; (x)id (x)5. Letv=Atu. Show that for allu2Rn:FP(u)=eihu;mi kvk2=26. Show the following:Theorem 132 Letn 2Mn(R)be a sym-metric and non-negative real matrix. There exists a unique complexmeasure onRn,denotedNn(m; ), with fourier transform:FNn(m; )(u)4=ZRneihu;xidNn(m; )(x)=eihu;mi 12hu; uifor ,Nn(m; )is a Probability 20: Gaussian Measures4De nition 142 Letn 2Mn(R)bea symmetric and non-negative real matrix. The Probability measureNn(m; )onRnde ned in theorem(132)is cal led then-dimensionalgaussian measureornormal distribution ,withmeanm2 Rnand covariance matrix .Exercise 1andm2Rn. Show thatNn(m;0) = 2Mn(R) be a symmetric andnon-negative real matrix. LetA2Mn(R) be such that = :Rn!Cis said to be apolynomial, if and only if, it is a nite linear complex combination of mapsx!x ,1for Show that for allB2B(R), we have:N1(0;1)(B)=1p2 ZBe x2=2dx1 See de nition (140).

3 20: Gaussian Measures52. Show that:Z+1 1jxjdN1(0;1)(x)<+13. Show that for all integerk 1:1p2 Z+10xk+1e x2=2dx=kp2 Z+10xk 1e x2=2dx4. Show that for all integerk 0:Z+1 1jxjkdN1(0;1)(x)<+15. Show that for all 2Nn:ZRnjx jdN1(0;1) ::: N1(0;1)(x)<+ 20: Gaussian Measures66. Letp:Rn!Cbe a polynomial. Show that:ZRnjp(x)jdN1(0;1) ::: N1(0;1)(x)<+17. Let :Rn!Rnbe de ned by (x)=Ax+m. Explain whythe image measure (N1(0;1) ::: N1(0;1)) is well-de Show that (N1(0;1) ::: N1(0;1)) =Nn(m; ).9. Show if 2 Nnandj j=1,thenx! (x) is a Show that if 02 Nnandj 0j=k+1, then (x) 0= (x) (x) for some ; 2 Nnsuch thatj j=kandj j= Show that the product of two polynomials is a Show that for all 2Nn,x! (x) is a Show that for all 2Nn:ZRnj (x) jdN1(0;1) ::: N1(0;1)(x)<+ 20: Gaussian Measures714. Show the following:Theorem 133 Letn 2Mn(R)be a sym-metric and non-negative real matrix. Then, for all 2Nn,themapx!x is integrable with respect to the Gaussian measureNn(m; ):ZRnjx jdNn(m; )(x)<+1 Exercise =( ij)2Mn(R) be a symmetricand non-negative real matrix.

4 Letj; be the fouriertransform of the Gaussian measureNn(m; ), :8u2Rn; (u)4=eihu;mi 12hu; ui1. Show that:ZRnxjdNn(m; )(x)=i 1@ 20: Gaussian Measures82. Show that:ZRnxjdNn(m; )(x)=mj3. Show that:ZRnxjxkdNn(m; )(x)=i 2@2 @ Show that:ZRnxjxkdNn(m; )(x)= jk+mjmk5. Show that:ZRn(xj mj)(xk mk)dNn(m; )(x)= 20: Gaussian Measures9 Theorem 134 Letn =( ij)2Mn(R)be a symmetric and non-negative real matrix. LetNn(m; )be thegaussian measure with meanmand covariance matrix .Then,forallj;k2Nn, we have:ZRnxjdNn(m; )(x)=mjand:ZRn(xj mj)(xk mk)dNn(m; )(x)= jkDe nition 143 Letn ( ;F;P)be a Probability space. LetX:( ;F)!(Rn;B(Rn))be a measurable map. We say thatXis ann-dimensionalgaussianornormal vector, if and only if itsdistribution is a Gaussian measure , (P)=Nn(m; )for somem2 Rnand 2Mn(R)symmetric and non-negative real the 20: Gaussian Measures10 Theorem 135 Letn ( ;F;P)be a Probability space. LetX:( ;F)!Rnbe a measurable map.

5 ThenXis a Gaussian vector,if and only if there existm2 Rnand 2Mn(R)symmetric andnon-negative real matrix, such that:8u2Rn;E[eihu;Xi]=eihu;mi 12hu; uiwhereh ; iis the usual inner-product nition 144 LetX:( ;F)! R(orC) be a random variableon a Probability space( ;F;P). We say thatXisintegrable,ifandonly if we haveE[jXj]<+1. We say thatXissquare-integrable,if and only if we haveE[jXj2]<+ to de nition (144), ShowXis integrable if and only ifX2L1C( ;F;P).2. ShowXis square-integrable, if and only ifX2L2C( ;F;P). 20: Gaussian Measures11 Exercise to de nition (144), supposeXis Show thatXis integrable, if and only ifXisP-almost surelyequal to an element ofL1R( ;F;P).2. Show thatXis square-integrable, if and only ifXisP-almostsurely equal to an element ofL2R( ;F;P).Exercise ;Y:( ;F)!(R;B(R)) be two square-integrablerandom variables on a Probability space ( ;F;P).1. Show that bothXandYare Show thatXYis integrable3. Show that (X E[X])(Y E[Y]) is a well-de ned and 20: Gaussian Measures12De nition 145 LetX;Y:( ;F)!

6 (R;B(R))be two square-integrable random variables on a Probability space( ;F;P).Wede- ne thecovariancebetweenXandY,denotedcov(X;Y ),as:cov(X;Y)4=E[(X E[X])(Y E[Y])]We say thatXandYareuncorrelatedif and only ifcov(X;Y)= ,cov(X;Y)is cal led thevarianceofX,denotedvar(X).Exercise ;Ybe two square integrable, real random variableon a Probability space ( ;F;P).1. Show thatcov(X;Y)=E[XY] E[X]E[Y].2. Show thatvar(X)=E[X2] E[X] Show thatvar(X+Y)=var(X)+2cov(X;Y)+var(Y)4. Show thatXandYare uncorrelated, if and only if:var(X+Y)=var(X)+var(Y) 20: Gaussian Measures13 Exercise ann-dimensional normal vector on someprobability space ( ;F;P), with lawNn(m; ), wherem2 Rnand =( ij)2Mn(R) is a symmetric and non-negative real Show that each coordinateXj:( ;F)!Ris Show thatE[jX j]<+1for all Show that for allj=1;:::;n,wehaveE[Xj]= Show that for allj;k=1;:::;n,wehavecov(Xj;Xk)= 136 LetXbe ann-dimensional normal vector on a prob-ability space( ;F;P),withlawNn(m; ).

7 Then,forall 2Nn,X is integrable. Moreover, for allj;k2Nn, we have:E[Xj]=mjand:cov(Xj;Xk)= jkwhere( ij)= . 20: Gaussian Measures14 Exercise the following:Theorem 137 LetX:( ;F)!(R;B(R))be a real random vari-able on a Probability space( ;F;P).Then,Xis a normal randomvariable, if and only if it is square integrable, and:8u2R;E[eiuX]=eiuE[X] 12u2var(X)Exercise ann-dimensional normal vector on a prob-ability space ( ;F;P), with lawNn(m; ). LetA2Md;n(R)beand nreal matrix, (n;d 1). Letb2 RdandY=AX+ Show thatY:( ;F)!(Rd;B(Rd)) is Show that the law ofYisNd(Am+b;A: :At)3. Conclude thatYis anRd-valued normal random 20: Gaussian Measures15 Theorem 138 LetXbe ann-dimensional normal vector with lawNn(m; )on a Probability space( ;F;P),(n 1).Letd 1andA2Md;n(R)be and nreal matrix. ,Y=AX+bis and-dimensional normal vector, with law:Y(P)=Nd(Am+b;A: :At)Exercise :( ;F)!(Rn;B(Rn)) be a measurable map,where ( ;F;P) is a Probability space. Show that ifXis a gaussianvector, then for allu2Rn,hu;Xiis a normal random :( ;F)!

8 (Rn;B(Rn)) be a measurable map,where ( ;F;P) is a Probability space. We assume that for allu2Rn,hu;Xiis a normal random Show that for allj=1;:::;n,Xjis Show that for allj=1;:::;n,Xjis square Explain why givenj;k=1;:::;n,cov(Xj;Xk) is well-de 20: Gaussian Measures164. Letm2 Rnbe de ned bymj=E[Xj], andu2Rn. Show:E[hu;Xi]=hu;mi5. Let = (cov(Xi;Xj)). Show that for allu2Rn,wehave:var(hu;Xi)=hu; ui6. Show that is a symmetric and non-negativen nreal Show that for allu2Rn:E[eihu;Xi]=eiE[hu;Xi] 12var(hu;Xi)8. Show that for allu2Rn:E[eihu;Xi]=eihu;mi 12hu; ui9. Show thatXis a normal Show the 20: Gaussian Measures17 Theorem 139 LetX:( ;F)!(Rn;B(Rn))be a measurable mapon a Probability space( ;F;P).Then,Xis ann-dimensional normalvector, if and only if, any linear combination of its coordinates is itselfnormal, or in other wordshu;Xiis normal for ( ;F)=(R2;B(R2)) and be the probabilityon (R;B(R)) de ned by =12( 0+ 1). LetP=N1(0;1) ,andX;Y:( ;F)!

9 (R;B(R)) be the canonical projections de ned byX(x;y)=xandY(x;y)= Show thatPis a Probability measure on ( ;F).2. Explain whyXandYare Show thatXhas the distributionN1(0;1).4. Show thatP(fY=0g)=P(fY=1g)= Show thatP(X;Y)= 20: Gaussian Measures186. Show for all :(R2;B(R2))!Cmeasurable and bounded:E[ (X;Y)] =12(E[ (X;0)] +E[ (X;1)])7. LetX1=XandX2be de ned as:X24=X1fY=0g X1fY=1gShow thatE[eiuX2]=e u2=2for Show thatX1(P)=X2(P)=N1(0;1).9. Explain whycov(X1;X2) is well-de Show thatX1andX2are LetZ=12(X1+X2). Show that:8u2R;E[eiuZ]=12(1 +e u2=2) 20: Gaussian Measures1912. Show thatZcannot be Conclude that althoughX1;X2are normally distributed, (andeven uncorrelated), (X1;X2) is not a Gaussian 2Mn(R)beasymmetric and non-negative real matrix. LetA2Mn(R)besuchthat =A:At. We assume that is non-singular. We de nepm; :Rn!R+by:8x2Rn;pm; (x)4=1(2 )n2pdet( )e 12hx m; 1(x m)i1. Explain why det( )> Explain whypdet( ) =jdet(A) Explain whyAis 20: Gaussian Measures204.

10 Let :Rn!Rnbe de ned by:8x2Rn; (x)4=A 1(x m)Show that for allx2Rn,hx m; 1(x m)i=k (x) Show that is aC1-di Show that (dx)=jdet(A) Show that:ZRnpm; (x)dx=18. Let =Rpm; dx. Show that:8u2Rn;F (u)=1(2 )n2 ZRneihu;Ax+mi kxk2=2dx9. Show that the fourier transform of is therefore given by:8u2Rn;F (u)=eihu;mi 12hu; 20: Gaussian Measures2110. Show that =Nn(m; ).11. Show thatNn(m; )<< dx, thatNn(m; ) is absolutelycontinuous to the Lebesgue measure 2Mn(R)beasym-metric and non-negative real matrix. We assume that is such that u=0andu6= 0. We de ne:B4=fx2Rn;hu;xi=hu;migGivena2Rn,let a:Rn!Rnbe the translation of ShowB= 1 m(u?), whereu?is the orthogonal Show thatB2B(Rn).3. Explain whydx(u?) = 0. Is it important to haveu6=0?4. Show thatdx(B)= 20: Gaussian Measures225. Show that :Rn!Rde ned by (x)=hu;xi, is Explain why (Nn(m; )) is a well-de ned Probability Show that for all 2R,wehave:F (Nn(m; ))( )=ZRnei hu;xidNn(m; )(x)8.