Multi-dimensional Central Limit Argument

Transcription

1 Mult-dmensonal Central Lmt Argument Outlne t as Consder d random proceses t, t,. 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 jontly Gaussan for any and for any samplng nstants. t

2 Gaussan Random Vector Jont Characterstc Functon Defnton. For a -dmensonal random vector,,,, defne ts jont characterstc functon as jx jx j,,, x,,, e e e f x x x dxdxdx where f x, x,, x s the jont pdf of. Usng expectaton notaton, j,,, e ( g) When the random varables are statstcally ndependent, j j j,,, e e e In the one-dmensonal case, j e Usng matrx notaton, Let ω,,, and,,,. hen ω and eq. g s wrtten as jω ω e g

3 3 Covarance Matrx Consder a -dmensonal random vector,,,. Defne cov(, ) j j j j and s called the covarance matrx of the random vector. When s a zero mean random vector, that s, 0 for,,,, In that case, j j ote that

4 4 Covarance matrx of the sum vector Let,,, be a zero-mean -dmensonal random vector. Let denote the covarance matrx of :. Consder ndependent vectors,,, statstcally dentcal to. Defne the sum vector as hen. g3 proof: Snce s a zero mean random vector, whch s j j j notng j j 0 for j

5 5 Jont Characterstc Functon of the sum vector Let and assume are d to. hen ln ω ω ln ( g4) proof: jω e exp j ω ω exp jω ω exp j notng are ndependent ω exp j ω ω notng are dentcal to

6 6 Moments Recall ω,,, and,,,. ω and ωe. Assume s a zero mean random vector. jω Defne the random varable W as W jω j hen ωe W W W3 W m! 3! nd term: W j We are assumng 0 and thus W 0. j 3rd term: W j j j j recallng covarance when 0 j j j ωλ ω j ωλ ω,,, j j m

7 7 Mult-dmensonal Central Lmt hm Let and be d to, where = 0. hen where lm ω exp ωλω ( m3) Λ Λ. s referred to as a zero-mean Gaussan random vector when ts jont characterstc functon s the form shown n eq.m3. proof: From eq.m and m, ω W W W3 3! 3! ωλ ω ω ln ln ωλ f 3 3 ω 3 u u Recallng ln( u) u ; u 3 ω ln ωλω 3 f3 other terms From eq.g4, ln ω ω ln ωλ ω f3 other terms Fnally lm ln and from eq.g3,. ω ωλω 3 f 3

8 8 Jont Char Functon of non-zero mean Gaussan Let be a Gaussan random vector wth mean m and covarance matrx Λ. hen ω exp ωλω jωm A Gaussan random vector s completely defned by the mean and ts covarance matrx. Proof: Defne m. hen s a zero-mean Gaussan random vector, and t s easy to see Λ Λ. From eq.m3, ω exp ωλω. hus ω exp jω exp exp jω m jω exp jωm ω exp jωm exp ωλω jωm notng Λ exp ωλω jωm Λ

9 9 Formal Defnton of Gaussan Random Vector s a Gaussan random vector (or the component random varables are jontly Gaussan) f and only ts jont characterstc functon s v exp ωλω jωm where m s the mean and Λ s the covarance matrx. he pdf f f ( 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 as a transformaton of A b where dm, A s a matrx, and b s a -dmesonal constant vector. hen s also a Gaussan random vector. m Am b and Λ AΛ A ote. Sum of any Gaussan s Gaussan. he component random varables need not be ndependent. proof: ω exp jω exp exp jω A b jωa exp jωb ωa exp jωb exp ωλω ωm ωb exp j exp j ωωa ωaλ ωa jωam exp jωb notng j CD exp ω AΛ A ω ω m A b D C s Gaussan wth m m A b and Λ AΛ A

11 Mult dmensonal Central Lmt Argument Example =3 ( t) ( t); ( t) are ndependent random telegraph sgnals () t () t () t t 0 As t t t3 3, ( t) becomes a Gaussan random process. t ( ), t ( ), t ( ) are jontly Gaussan. 3

12 Covarance Matrx of a Random elegraph Sgnal t ( ) s a random telegraph sgnal wth transton rate [transtons/second] We have shown that t ( ) s WSS wth mean m ( t) 0 ; t t varance () () ; auto-correlaton, R ( ) e We tae 3 tme samples. t t t Samplng tme nstants are,, (,,3) seconds 3 3,,,, 3 s a 3-dmensonal random vector. 0 for,,3 or n vector notaton, the mean value vector m 0. cov(, ). j j Snce 0, j R ( t t) e j j j t Let be the covarance matrc of the random vector. hen 4 e e e e 4 e e j t Covarance matrx of the sum vector Defne. hen.

13 3 Jont Characterstc Functon 3,, a 3-dmensonal random vector jx jx j3x 3,,,, 3 e e e f x x x3 dxdxdx 3 3 where f x, x, x s the jont pdf of. Usng expectaton notaton, j33,, e e 3 Usng matrx notaton, Let ω,, and,,. hen ω 3 3 ωe jω 3 3 and eq.e s wrtten as For the sum vector, 3,, a 3-dmensonal random vector jz jz j3z 3,,,, 3 e e e f z z z3 dzdzdz 3 3 where f z, z, z s the jont pdf of. Usng matrx notaton, e jω ω We do not now ω yet. However we can fnd ω wthout nowledge of ω.

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

15 5 Example =. t t Let the samplng tme nstants be, (,) seconds. For the random telegraph sgnal,,, s a -dmensonal random vector. 0 for, or n vector notaton, the mean value vector m 0. he covarance matrx s Λ e e. For the sum vector, wth the covarance matrx As, ω exp ωλω ( e) e where Λ Λ e Eq.e s the jont characterstc functon of a zero-mean Gaussan random vector.

16 6 he correlaton coeffcent between the random varables and s Wrte hen and Λ ωλ ω VAR cov, VAR. e exp ω he jont pdf s f () z exp zλ z Λ where Λ f Λ () z exp zλ z Λ z z exp, z zzz z z exp b

17 7 Comparng wth our prevous defnton of the jontly Gaussan random varables f ( x, y) e, / ( ) x x y y ( ) Eq.b s the pdf of the jontly Gaussan random varables and wth 0 VAR( ) VAR( ), Homewor due by ov 9, 0 Derve the autocorrelaton functon of the random process ( t).

18 8 Gaussan Random Process ( t) s referred to as a Gaussan random process f t, t,, t are jontly Gaussan for any samplng nstants t, t,, t and for any number of samples. Let,,,. t t t ω exp ωλω jωm where m s the mean and Λ s the covarance matrx. s referred to as a Gaussan random vector (or the component random varables are jontly Gaussan) f and only ts jont characterstc functon s he jont pdf f ( x) s completely defned by the mean m and the covarance matrx Λ f ( x) exp xm Λ xm Λ Statonary Gaussan Random Process A wde-sense staonary Gaussan random process s a staonary Gaussan random process. o show t s statonary, we mush show the pdf s tme-shft-nvarant. f ( x x,, x ) f ( x x,, x ),,,,,,,, t t t t t t for any, any choce of t t t, and for any. However, when the process s Gaussan, the pdf s completely defned by the mean vector m and the covarance matrx Λ. So f we show m and Λ are tme-shft-nvarant, then t s statonary. m,,, If ( t) s wde-sense statonary, then t m, fxed, for any samplng nstant. So m t t t s tme-shft-nvarant. j

19 9 t t R t t t t cov,, j j j j If ( t) s wde-sense statonary, then R t, t R t t and thus s tme-shft-nvarant. j j j Is the Sum Process statonary Gaussan? When ( t) s WSS, the sum process ( t) s a statonary Gaussan random process. When ( t) s WSS, Λ s tme-shft nvarant, that s, Λ Snce Λ depends only on the dfference between samplng tmes. Λ, t ( ) s WSS. However snce t ( ) s Gaussan, t ( ) s a statonary Gaussan random process.

20 0 Random Process through Lnear Flter () t t () ht () Gaussan Input/Output he output of a lnear flter s Gaussan when the nput s Gaussan. t () ( ) ht ( ) d For any tme sample t t, t ( ) ( ) ht ( ) d ( ) h( t ) ( ) are jontly Gaussan ramdom varables. t ( ) s a weghted sum of jontly Gaussan random varables. Applyng the above argument to multple samples, t ( ) are jontly Gaussan random varables So t ( ) s a Gaussan random process.

21 Moment of the Output For any nput process ( t), m () t m ( ) h( t ) d R (, ts) R (, ) ht ( ) hs ( ) dd ( w) t () ( ) ht ( ) d m () t () t ( ) h( t ) d ( ) h( t ) ( )( h t ) ( ) h( t ) d m ( ) h( t ) d R (, t s) () t ( s) ( ) ht ( ) d ( ) hs ( ) d ( ) ( ) h( t ) h( s) dd R (, ) h( t ) h( s) dd

22 Wde Sense Statonary Input/Output he output of a lnear flter s WSS when the nput s WSS. From eq.w, m () t m ( ) h( t ) d When t ( ) s WSS, m ( ) mand thus m () t m h( ) d mh(0) m ( t) does not vary wth t. R (, t s) R (, ) h( t ) h( s) dd When t ( ) s WSS, R (, ) R ( ) and thus R (, ts) R ( ) ht ( ) hs ( ) dd R ( t, s) depends only on s t. changng varables u t and vs, R ( uvst) h( u) h( v) dudv