By M. Frank Norman (Eds.)

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Example text

Since P(X) is complete with respect to A, there is a K" (x) E P ( X ) to which R " ( x ) converges weakly. 12) for each f E L ( X ) and x E X . ,B) E B(X) for each B E W,so that K m is a stochastic kernel. We omit the proof. ) Clearly uu" = 8 " =~8" +- n-'U" - n-11. 1 1) follows. If U is ergodic, U " f ( x ) = U m f ( y ) for all f E L(X), so that K"(x, . ) for all x , y E X . ) is stationary. f) + as n - t co, for all f E (P, U " f ) = U"f = ( K " , f ) L(X). Hence p = K". I The next result gives a rather special property of Doeblin-Fortet operators.

The metric space ( X , d ) is obviously compact, and B ( X ) = L ( X ) contains all complex valued functions on X. Clearly (1 f 1 < RIfl, where R = 1 + 2/mind(x,x').

A useful representation of the powers of the transition operator w - ( x )= = (eo, .. where e" . 7) and u ( x , e") is defined iteratively: u(u(x,e"), en) = u ( x , e n + ' ) . 7) makes sense and defines a measurable function of x. For any A E g'", P ~ ( X , S - ~=A P) x ( L f NA~) = = E x ( P ( s N €AIX,)) Ex(Pm(xN, A ) ) = U N p m( x ,A ) . 1 yields additional information about event sequences in distance diminishing and finite state models. The Markov process X: = (X,,, En) can be used in the same way for finite state models.

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