Multi-dimensional Central Limit Theorem

Transcription

1 Mult-dmensonal Central Lmt heorem Outlne ( ( ( t as ( + ( + + ( ( ( Consder a sequence of ndependent random proceses t, t, dentcal to some ( t. Assume t = 0. Defne the sum process t t t t = ( t = (; t ( t are d to (, t ( t = 0 = { } ( t tme ( t ( t t t As, ( t becomes a Gaussan random process. t, t,, t are ontly Gaussan for any and for any samplng nstants. { } t

2 Jont Characterstc Functon of a Random Vector defne ts ont characterstc functon as ( Defnton. For a -dmensonal random vector =,,,, ωx ωx ω (,,, x Φ ω ω ω (,,, e e e f x x x dxdxdx where f x, x,, x s the ont pdf of. Usng the expectaton notaton, ( ω+ ω+ ω Φ ( ω, ω,, ω = e ( g When the random varables { } are statstcally ndependent, ( ω ω ω ω ω,,, Φ = e e e ω In the one-dmensonal case, Φ ω = e ω Our vectors are row vectors. Usng matrx notaton, ( ω ω ω ( Let ω =,,, and =,,,. hen ω and eq. g s wrtten as = ω + ω + + ω ω e ( g Φ = ω

3 Covarance Matrx of a Random Vector Consder a -dmensonal random vector,,,. Defne and = λ = cov(, = λ λ λ λ λ λ Λ = λ λ λ Λ s referred to as the covarance matrx of the random vector. When s a zero mean random vector, that s, = 0 for every =,,,, In that case, ote that λ = Λ = ( = =

4 4 Covarance Matrx of the Sum Vector ( Let =,,, be a zero-mean -dmensonal random vector. Let Λ denote the covarance matrx of : Λ =. Consder ndependent vectors,,, statstcally dentcal to. Defne the sum vector as =. = hen Λ = Λ ( g ( t ( t = (;{ (} t t are d to (, t ( t = 0 = tme ( t ( t t t As, ( t becomes a Gaussan random process. { t, t,, t ( } are ontly Gaussan for any and for any samplng nstants. t Proof Snce s a zero mean random vector, Λ = ( ( whch s = = = = = = = + notng = = 0 for Λ = = = Λ =Λ =

5 5 Jont Characterstc Functon of the Sum Vector Let = and assume { } are d to. = hen ln ω Φ ω = ln Φ ( g4 Proof Φ ( ω = e ω = e ω = = = e ω notng { } are ndependent = = e ω = ω = Φ = Φ ω notng { } are dentcal to

6 6 Jont CF of a zero-mean random vector Recall, for a -dm random vector, ts ont characterstc functon s where Φ ( ω = e ( ω ω ω ( ω =,,, and =,,,. ω ω = ω + ω + + ω Defne the random varable W as ( ω ω ω W = ω = hen Φ ( ω = e W W W = + W + + +!! ( m ow assume s a zero-mean random vector. he nd term of eq. m s ( ω ω ω W = However, for a zero-mean vector, = 0 and thus W = 0. rd term of eq.m: ( ω ω ω W = ( ω ω ω ( ω ω ω = = = = ω ω recallng the covarance λ = when = = 0 = = = = ωλ ω ωλ ω ( m

7 7 Mult-dmensonal Central Lmt heorem Let = and { } be d to, where = 0. hen where = ωλ ω lm Φ ( ω = e ( m Λ = Λ. s referred to as a zero-mean Gaussan random vector when ts ont characterstc functon s the form shown n eq.m. Proof From eq. m and m, ω W W W Φ = !! = ωλ ω + ω ln Φ = ln ωλ + f ω u u Recallng ln( + u = u + ; u < ω ln Φ = ωλω + f + other terms From eq. g4, f Fnally ω ln Φ ω = ln Φ = ωλω + f + other terms lm ln Φ and from eq. g, Λ = Λ. ω = ωλω

8 8 Jont Char Functon of non-zero mean Gaussan Let be a Gaussan random vector wth mean m and covarance matrx Λ. hen ts ont CF s ωλ ω Φ =e ω + ωm A Gaussan random vector s completely defned by the mean and ts covarance matrx. Proof. Defne Y = m. hen Y s a zero-mean Gaussan random vector, and t s easy to see Λ = Λ. Y From eq.m, Φ Y ω = exp ωλyω. hus ω exp ( ω Φ = ( ω( Y m = exp + = exp =Φ Y ( ωy exp ( ωm ω exp ( ωm = exp ωλyω + ωm = exp ωλω + ωm notng Λ = Λ Y

9 9 Formal Defnton of Gaussan Random Vector s a Gaussan random vector (or the component random varables are ontly Gaussan f and only f ts ont characterstc functon s Φ ω = exp ωλω + ωm where m s the mean vector and Λ s the covarance matrx. he ont pdf f f ( π ( x can be found by the nverse Fourer transform: ( ( x = exp x m Λ x m Λ

10 0 Weghted Sum of Gaussan Random Varables Let be a Gaussan random vector and defne Y as a transformaton of Y = A + b where dm =, A s a matrx, and b s a -dmesonal constant vector. hen Y s also a Gaussan random vector wth m Y and Y = Am + b Λ = AΛ A ote. A sum of Gaussan random varables s Gaussan. he component Gaussan random varables { } don't have to be ndependent for the sum to be Gaussan. Homewor. Weghted Sum of Gaussan Random Varables Prove that a transformaton of a Gaussan random vector s a Gaussan random vector. Hnt. ω( AΛ A ω ω m A b Show Φ ω =e + + to prove Y s Gaussan wth Y Y = + and Y = m m A b Λ AΛ A

11 Mult-dmensonal Central Lmt hm - Example = ( t = ( t; { ( t } are ndependent random telegraph sgnals = ( t ( t ( t t = 0 As t = t = t =, ( t becomes a Gaussan random process. t, t, t are ontly Gaussan. { }

12 Covarance Matrx of the Random elegraph Sgnal Samples ( t s a random telegraph sgnal wth transton rate α [transtons/second] We have shown that ( t s a WSS random process wth mean m ( t = 0 ; t t varance σ ( = ( = ; auto-correlaton, R ( τ = e ατ In ths example, we wll tae = tme samples. ( t t t = [ ] Samplng tme nstants are,, (,, seconds. ( ( =,, =,, s a -dmensonal random vector. = 0 for =,, or n vector notaton, the mean vector m = 0. λ = cov(,. Snce = = 0, λ = = R ( t t = e α t Let Λ be the covarance matrc of the random vector. hen α 4α e e α α Λ e e = 4α α e e t Covarance Matrx of the Sum Vector Defne =. hen Λ =Λ. =

13 Jont Characterstc Functon = (,, a -dmensonal random vector e e e f x x x dxdxdx ωx ωx ωx Φ ( ω, ω, ω (,, where f x, x, x s the ont pdf of. Usng expectaton notaton, ( ω+ ω+ ω Φ,, = e e ( ω ω ω Usng matrx notaton, ( ω ω ω ( Let ω =,, and =,,. hen ω = ω + ω + ω Φ ( ω = e ω eq. e and s wrtten as For the sum vector, = (,, a -dmensonal random vector e e e f z z z dzdzdz ωz ωz ωz Φ ( ω, ω, ω (,, where f z, z, z s the ont pdf of. Usng matrx notaton, Φ = e ω ω We do not now Φ ( ω yet. However, as, we can fnd Φ ( ω wthout nowledge of Φ ( ω.

14 4 Mult-dmensonal Central Lmt heorem As, Φ ω = exp ωλω ( e α 4α e e α α where Λ e e = Λ = 4α α e e Eq. e s the ont characterstc functon of a zero-mean Gaussan random vector. Jont pdf of the Gaussan Random Vector he ont pdf f ( z can be found by the nverse Fourer transform from Φ ( ω : f ( π ( ( z = exp zλ z wth =. Λ