16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde Lecture 4 Last time: Left off with characteristic function 4. Prove (t)=lo (n)where X=x,+x2++Xn(X; independent Let S=X,+X,+.X, where the X; are independent d, (=EJe=Een(x+x ++) ELe ] E[e:]E[lek This is the main reason why use of the characteristic function is convenient This would also follow from the more devious reasoning of the density function for the sum of n independent random variables being the nth order convolution of the individual density functions- and the knowledge that convolution in the direct variable domain becomes multiplication in the transform domain 5. MacLaurin series expansion of p(n) Becausef'x) is non-negative and /(xdx=(or, even better, JI/(x]dx follows that JI/(x)dx=l converges so that f(ax)is Fourier transformable. Thus the characteristic function o() exists for all distributions and the inverse relation p(0)f(r)holds for all distributions. This implies that p(r) is analytic for all real values of t Then it can be expanded in a power series, which converges for all finite values d)=O)+p(0)1+,p2(0)2+…+-p(0)+ 0(1)=f(x)edx,c(0)=1 Page 1 of 616.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Lecture 4 Last time: Left off with characteristic function. 4. Prove φ () = Πφ i () where X = X1 + X 2 + ... + X (Xi t t independent) x x n Let S X = 1 + X + ...X where the Xi are independent. 2 n ... t ⎤ jtS ⎤ = E ⎡e jt X 1 +X 2 + +Xn ) φ () = E ⎣ ⎡e ⎦ ⎣ ( ⎣ s ⎦ jtX1 ⎤ Ee jtX jtXn ⎡ ⎡ ⎤ ⎣ ⎦ = Ee ⎦ ⎣ ⎡ 2 ⎤ ⎦...Ee = ∏φ ( )t Xi This is the main reason why use of the characteristic function is convenient. This would also follow from the more devious reasoning of the density function for the sum of n independent random variables being the nth order convolution of the individual density functions – and the knowledge that convolution in the direct variable domain becomes multiplication in the transform domain. 5. MacLaurin series expansion of φ( )t ∞ ∞ Because f(x) is non-negative and f ( ) x dx = 1 (or, even better, f ( ) x dx = 1), it ∫ ∫ −∞ −∞ ∞ follows that ∫ f ( ) x dx = 1converges so that f(x) is Fourier transformable. Thus −∞ the characteristic function φ( )t exists for all distributions and the inverse relation φ() t → f (x) holds for all distributions. This implies that φ( )t is analytic for all real values of t. Then it can be expanded in a power series, which converges for all finite values of t. 1 2 0 2 n n φ(t) = φ(0) +φ 0 0 ( ) ( )t + 1 φ ( ) ( )t + ... + 1 φ ( ) ( )t + ... 2! n! ∞ jtx φ() t = ) dx , ∫ f (x e φ(0) = 1 −∞ Page 1 of 6