Multi-dimensional Central Limit Theorem

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ÊΝηρεύς Ζάππας
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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 tme () t () t t t t () t As, becomes a Gaussan random process, that s to say, { } t, t,, t are jontly Gaussan for any and any samplng nstants.

2 Jont Characterstc Functon of a Random Vector defne ts jont characterstc functon as ( ) Defnton. For a -dmensonal random vector,,,, jωx jωx jω (,,, x Φ ω ω ω ) (,,, ) e e e f x x x dxdxdx where f x, x,, x s the jont pdf of. Usng the expectaton notaton, j( ω+ ω+ + ω Φ ) ( ω, ω,, ω ) e ( g) When the random varables { } are statstcally ndependent, j j ( ω ω ω ) ω ω j,,, Φ e e e ω In the one-dmensonal case, j 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 j e ( g) Φ ω ω

3 3 Jont Characterstc Functon of a Subset ( ) Let,,,. ( ) ( ) ( ) Consder a subset of the random varables, say,,,,, <. he jont characterstc functon of,,, can be found easly from the jont char functon of,,, : Φ,,, Φ ( ω, ω,, ω ) (,,, ) e e e f,,, x x x dxdx dx + jω x jω x jω x jω x jω x jω x (,,,,,, ) e e e f,,,,,, x x x x x dxdx dx dx dx + + +,,,,,, ( ω, ω,, ω, ω+ 0,, ω 0) Example ( ω, ω ) ( ω, ω 0, ω ) Φ Φ, 3,, ( ω ) ( ω, ω 0, ω 0) Φ Φ,, 3 3

4 4 Covarance Matrx of a Random Vector Consder a -dmensonal random vector,,,. Defne and λ cov(, ) j j j j λ λ λ λ λ λ λ λ λ s referred to as the covarance matrx of the random vector. property he dagonal elements of the covarance matrx are σ λ cov, j,,, jj j j j that s, the varance of. j property For j, that s, λ cov, λ j j j j j s symmetrcal. property he correlaton coeffcent s cov (, ) j λj ρj σσ σσ j j hus λ ρ σσ j j j

5 5 property When s a zero mean random vector, that s, 0 for every,,,, In that case, ote that λ j j ( )

6 6 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) () t ();{ ()} t t are d to (), t () t 0 () t tme () t () t Proof Snce s a zero mean random vector, t t As, ( t) becomes a Gaussan random process. { t, t,, t ( )} are jontly Gaussan for any and for any samplng nstants. t ( )( ) whch s j j j notng j j 0 for j

7 7 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 jω e jω e j ω notng { } are ndependent e j ω ω Φ Φ ω notng { } are dentcal to

8 8 Jont CF of a zero-mean random vector For a -dm random vector where Φ ( ω) e ( ω ω ω ) ( ) jω, ts jont characterstc functon s ω,,, and,,,. ω ω + ω + + ω Defne the random varable W as hen Φ ( ω) e W ( ω ω ω ) W jω j W W3 + W + + +! 3! ow assume s a zero-mean random vector. he nd term of eq. m s ( ω ω ω ) W j For a zero-mean vector, 0 for all j. hus we have j ( m ) ( m ) W 0 he 3rd term of eq. m s ( ω ω ω ) W j ( ω ω ω )( ω ω ω ) j ω ω j j recallng the covarance λ when 0 j j j j ωλ ω ω ω j j ( m3)

9 9 Mult-dmensonal Central Lmt heorem Let and { } be d to. Assume 0. hen ω ω lm Φ ( ω) e ( m4) where. s referred to as a zero-mean Gaussan random vector when ts jont characterstc functon s the form shown n eq. m4. Proof From eq. m, m, and m3, 3 ω W W W Φ ! 3! ω ω + ω ln Φ ln ω + f 3 3 ω 3 u u Recallng ln( + u) u + ; u < 3 ω ln Φ ωω + f other terms From eq. g4, 3 f 3 Fnally ω ln Φ ω ln Φ ωω + f 3 + other terms lm ln Φ ( ω) ωω and from eq. g3,.

10 0 Jont Char Functon of non-zero mean Gaussan Let be a Gaussan random vector wth mean m and covarance matrx. hen ts jont CF s ω ω Φ e ω + jωm 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. Y From eq. m4, Φ Y ω exp ωyω. hus ω exp ( jω ) Φ ( jω( Y m ) ) exp + exp Φ Y ( jωy ) exp ( jωm ) ω exp ( jωm ) exp ωyω + jωm exp ωω + jωm notng Y

11 Formal Defnton of Gaussan Random Vector s a Gaussan random vector (or the component random varables are jontly Gaussan) f and only f ts jont characterstc functon s Φ ω exp ωω + jωm where m s the mean vector and s the covarance matrx. he jont pdf f f ( π ) ( x) can be found by the nverse Fourer transform: ( ) ( x) exp x m x m Example For, m [ μ] [ ] σ Φ ω exp ωσ ω+ jωμ

12 For, m ( ) [ μ μ ] jontly Gaussan wth correlaton coeffcent ρ ( ) λ σ λ cov, λ cov (, ) λ σ ( ) cov, recallng ρ σσ σ ρσσ ρσσ σ Φ ω exp ωω + jωm ω ω ω [ ω ω ] ωm σ ρσσ ω [ ωω] ρσ ω σ σ ω σ + ωω ρσ σ + ω σ [ ω ω ] μ μ ωμ + ω μ ω σ + ωω ρσ σ + ω σ Φ ω + + exp j ( ωμ ω μ )

13 3 he jont pdf of a Gaussan random vector s For, x m ( ) ( x x ) f,.,. ( μ μ ) ( x μ x μ ) ( ) ( x) exp x m x m,. x m,. σ ρσσ. ρσσ σ σ σ ρ σ σ σ σ ρ. ( π ) σ ρσ σ ( x m ) ( x m ) ρσσ. σ, / πσσ ( ρ ) ( x ) ( x )( x ) + ( x ) σ μ ρσσ μ μ σ μ σσ Fnally we have, for, f ( x, x ) e ( ρ ) ( ρ ) x μ x μ x μ x μ ρ + σ σ σ σ μ μ μ μ x x x x ρ + σ σ σ σ ( ρ )

14 4 Weghted Sum of Gaussan Random Varables Let be a Gaussan random vector and defne Y as a lnear 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 Y + Y ( w ) m Am b and A 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( A ) A ω jω m A b Show Φ ω e + + to prove Y s Gaussan wth Y Y + and Y m m A b A A ote. A can be a h matrx wth h<. Eq. w stll holds true.

15 5 Example: Lnear Combnaton of Gaussan Random Varables: ( μ σ ) ( μ σ ) Let ~, and ~,. Suppose and are jontly Gaussan wth correlaton coeffcent ρ. Defne Y a + a Y ( Y Y ) Fnd the covarance matrx of,. Soluton. In ths example, a a A 0 b 0 Y and Y and are jonly Gaussan wth m Y a a μ aμ+ aμ Am 0 μ μ Y a a σ ρσσ a AA 0 ρσ a σ σ 0 a a a a a a σ + ρ σσ + σ σ + ρ σσ aσ + ρaσσ σ ( + ) σ + ρ σσ + σ Covarance matrx shows VAR Y a a a a. Y Alternate method of fndng VAR Y : VAR Y VAR a a (, ) VAR a + VAR a + COV a a ( ) σ σ a + a + aa a a σ σ a + a + a a σ σ, a + a + a a COV σ σ ρσσ a + a + a a

16 6 Suppose we are nterested n Y only. Let In ths example, Y a + a A [ a a ] he dmenson of Y can be smaller than that of. Y and s a Gaussan random varables wth m Y Covarance matrx μ Am [ a a ] a + a μ μ μ σ ρσσ a Y AA [ a a] ρσ a σ σ Y σ ρ σσ σ a + a a + a a σ s, and shows VAR Y + a a ρσ σ + a σ

17 7 Mult-dmensonal Central Lmt hm - Example 3 ( t) ( t); { ( t) } are ndependent random telegraph sgnals () t () t () t t 0 As t t t 3 3, ( t) becomes a Gaussan random process. t, t, t are jontly Gaussan. { } 3

18 8 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) ( t) 0 ; t t varance σ ; ατ auto-correlaton R ( τ ) e In ths example, we wll tae 3 tme samples. ( t t t ) [ ] Samplng tme nstants are,, (,,3) seconds. 3 ( ) ( () ),,,, 3 s a 3-dmensonal random vector. 3 0 for all,,3. he mean vector m 0. Let be the covarance matrc of the random vector. λ λ λ 3 λ λ λ3 λ 3 λ3 λ 33 where λ cov(, ). j j t j λj j R tj t e α Snce 0, We have α 4α e e α α e e 4α α e e j t

19 9 Covarance Matrx of the Sum Vector () () () Generate d random telegraph sgnals t, t,, t. Sample the sgnals to get d random vectors,,,. he sum vector s wth.. Jont Characterstc Functon ( ),, a 3-dmensonal random vector 3 3 e e e f x x x3 dxdxdx 3 jωx jωx jω3x Φ 3 ( ω, ω, ω ) (,, ) where f x, x, x s the jont pdf of. 3 Usng expectaton notaton, j( ω+ ω+ ω33 Φ,, e ) e ( ω ω ω ) 3 Usng matrx notaton, ( ω ω ω ) ( ) Let ω,, and,,. hen ω 3 3 ω + ω + ω Φ ( ω) e jω 3 3 eq. e and s wrtten as For the sum vector, ( ),, a 3-dmensonal random vector 3 3 e e e f z z z3 dzdzdz 3 jωz jωz jω3z Φ 3 ( ω, ω, ω ) (,, ) where f z, z, z s the jont pdf of. 3 Usng matrx notaton, Φ e jω ω We do not now Φ ( ω) yet. However, as, we can fnd Φ ( ω) wthout nowledge of Φ ( ω).

20 0 Mult-dmensonal central lmt theorem states As, Φ ω exp ωω ( e) α 4α e e α α where e e 4α α e e Eq. e s the jont characterstc functon of a zero-mean Gaussan random vector. Jont pdf of the sum vector he jont pdf f ( z) can be found by the nverse Fourer transform from Φ ( ω) : f ( π ) ( ) ( z) exp z z wth 3.

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Multi-dimensional Central Limit Theorem

Multi-dimensional Central Limit Theorem 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 ();