16.322 Stochastic Estimation and Control, Fall 2004 Prof vander velde [x()±3)=x(4)±2x(4)y(4)+y(2 =R(1,4)±2R2(1,2)+Rn(2)20 干R(44)≤[R2(4)+Bn(a2) R0,4)55[R2(4)+Rn(a (1)2+y(t2)2 k4)52x0)+x) Intuitively we feel that if the conditions under which an experiment is performed are time independent, then the statistical quantities associated with a random process resulting from the experiment should be independent of time analytically we say that a process is stationary if every translation in time transforms members of the process into other members of the process in such a way that probability is preserved This could also be stated by the statement that all distribution functions associated with the process (x1,1,x2,l2,xn,n)=F(x11,x2,41 be functions of the differences in the t, only and independent of the actual values of the 4. Define n-1 t. Then f(m is independent of t, f(x,1)→f(x) x(1)→x16.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 4 of 9 [ ]2 2 2 1 2 1 12 2 11 12 2 2 12 11 2 2 12 11 2 2 2 2 1 2 2 2 12 1 2 () ( ) () 2()( ) ( ) (,) 2 (, ) (, ) 0 1 (, ) (,) (, ) 2 1 (, ) (,) (, ) 2 1 () () 2 1 (, ) () ( ) 2 xx xy yy xy xx yy xy xx yy xx xt yt xt xt yt yt R tt R tt R tt R tt R tt R tt R tt R tt R tt xt yt R t t xt xt ± =± + =± + ≥ ≤ + ⎡ ⎤ ⎣ ⎦ ≤ + ⎡ ⎤ ⎣ ⎦ ≤ + ⎡ ⎤ ⎣ ⎦ ≤ + ⎡ ⎤ ⎣ m ⎦ Intuitively we feel that if the conditions under which an experiment is performed are time independent, then the statistical quantities associated with a random process resulting from the experiment should be independent of time. Analytically we say that a process is stationary if every translation in time transforms members of the process into other members of the process in such a way that probability is preserved. This could also be stated by the statement that all distribution functions associated with the process ( )( ) () () 11 2 2 11 21 2 1 , , , ,..., , , , , ,..., , n n F xtxt xt F xtxt xt nn n n = ++ τ τ be functions of the differences in the i t only and independent of the actual values of the i t . Define n −1 i τ . Then ( ) n f is independent of 1 t . 2 2 ( ,) ( ) ( ) ( ) f xt f x xt x xt x ⇒ ⇒ ⇒