Ref. Gallager, Stochastic Processes. Notation a vector. All vectors are row vectors. k k. jωx. Φ joint chacteristic function of.

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1 Gaussia Radom ector Ref. Gallager, Stochastic Processes. Notatio a vector. All vectors are row vectors a a aa ω matrix scalar matrix ( ) ( ) -dim radom vector,, -dim real vector ω,, ω Φ oit chacteristic fuctio of. Φ e ormalized IID Gaussia radom vector ω Normalized IID Gaussia Radom ector radom vector if { } ( ) Defiitio 3.3,,, is referred to as a -dimesoal ormalized IID Gaussia are ormalized idepedet Gaussia radom variables. Property. he oit pdf of a -dim ormalized IID Gaussia rvec is f ( v) ( ) ( v v ) where v,,. v v v vv f v i i e e ( π) ( π) Lemma 3.3. ( ω ω ω ) For ay -dim real vector ω,,,, { } cosider a liear combiatio of ormalized idepedet Gaussia radom variables : ω + ω + + ω ω he each term ω. is a idepedet 0,, ad thus is 0, N ω N ω N 0,. ωω ( ) ( )

2 Lemma 3.3. he oit CF of a ormalized IID Gaussia radom vector Φ e ωω is proof. he oit CF of is Φ e. ( s) Defie radom variable as ω. he Φ e he oe-dim characteristic fuctio of is Φ e, ad it ca be writte that ( s) ( g) Φ Φ s ω ( ) From lemma 3.3., is N 0, ωω, ad thus ( s) e ( g) Φ eq.g ad g prove the lemma ωω s s Joitly Gaussia Radom ariables Defiitio each { },,, is a set of zero-mea oitly Gaussia radom variables if ca be expressed as a liear combiatio of some fite set of ormalized idepedet Gaussia rvars {,,, }: m a,,,,. m Property. the each { } { } If,,, are zero-mea oitly Gaussia radom variables, is margially a zero-mea Gaussia radom variable. If,,, is a set of zero-mea oitly Gaussia rvars, each is a liear combiatio of a set of ormalized idepedet Gaussia rvars. a + a + + a. m m he sum of idepedet Gaussia rvars is a Gaussia rvar. is a zero-mea Gaussia rvar. he above defiitio ca be writte i a vector form as below

3 3 Defiitio. is a -dim zero-mea Gaussia radom vector if for some m-dim ormalized IID Gaussia radom vector, ca be expressed as [ ] where A a,, m is a give matrix of real umbers. { } ( ) Note. o say,,, are oitly Gaussia rvars is syoymous to say,,, is a Gaussia rvec. More geerally, { U U U } ( U U U ) Defiitio 3.3.,,, are oitly Gaussia radom variables, or equivaletly, U,,, is a -dim Gaussia radom vector if U + μ where is a -dim zero-mea Gaussia radom vector ad μ is a -dim real vector. ( ) ( Y Y Y ) [ B], hm Let,,, be a -dim zero-mea Gaussia radom vector, ad let Y,,, be a -dim radom vector satisfyig Y he Y is a zero-mea Gaussia radom vector. [ ][ ] [ B] [ B] [ A ] Sice is a zero-mea Gaussia rvec, for some ormalized IID Gaussa rvec. he Y B A is a m matrix ad thus Y is a zero-mea Gaussia radom vector. he above theorem ca be easily exteded to arbitrary mea.

4 4 ( ) Y [ B] ( ) hm 3.3..ext Let,,, be a -dim Gaussia radom vector, ad let Y Y, Y,, Y be a -dim radom vector satisfyig he Y is a Gaussia radom vector. [ B], ( μ ) [ B] [ B] [ B] [ C] m Sice is a Gaussia rvec, + μ for some ormalized IID Gaussia rvec ad some real vector μ. he Y herefore Y is a Gaussia rvec μ ( ) Property. If,,, is a Gaussia radom vector, the each is margially Gaussia. ( μ ) If is a Gaussia rvec, the for some ormalized IID Gaussia rvec ad for some vector μ, + μ he each a + a + + a + μ.. m m is N, a a a. m Property. It is ot correct to say that ucorrelated Gaussia radom variables are idepedet. "Ucorrelated oitly Gaussia radom variables are idepedet."

5 5 Lemma. For ay -dim zero mea radom vector ad ay -dim real vector ω, defie a radom variable ω. he ωλ ω where Λ is the covariace matrix of., ( ω ω ω ) ( ω ω ω )( ω ω ω ) i ω ω i i ωλ ω ( 3) hm Let be a zero-mea Gaussia radom vector with covariace matrix Λ. he the oit characteristic fuctio of is completely determied by Λ Φ e ωλ ω he oit characteristic fuctio of is Φ e. ω ( ω ) e e Φ ( s) ( ) s ( s) s ( ω ω ω ) Whe is a -dim zero-mea Gaussia rvec, for ay give -dim real vector ω,,,, ω is a zero-mea Gaussia rvar. Let ω. he Φ ( ) ω where Φ is the characteristic fuctio of with as the real parameter. Sice 0, Φ s e ad σ. herefore σ s ( ω ) e. ( ) Φ From the previous lemma, ωλ ω

6 6 ( ) Property. he oit CF Φ of a zero-mea Gaussia radom vector is completely determied by the covariace matrix Λ. O the other had, Φ determies the pdf of uiquely. herefore f z is also completely determied by the covariace matrix. Λ Lemma. Let U + μ for some zero-mea Gaussia radom vector ad for some real vector μ. he U ad have the same covariace matrix. cov( U, U ) U U U U ( )( ) + μ + μ + μ + μ ( )( ) + μ + μ μ μ Sice 0, cov( U, U ) cov(, ) hm ext Let U be a Gaussia radom vector with a arbitrary mea. Let U + μ for some zero-mea Gaussia radom vector. μ is the mea value vector. he the oit characteristic fuctio of U is completely determied by μ ad Λ. ωλω ωμ e Φ U U ad have the same covariace matrix Λ. Φ e U e ωu ( + μ) ω ωμ e Φ