Lectures on Prediction Theory: Delivered at the Univesity by K. Urbanik (auth.)

By K. Urbanik (auth.)

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SzegS, Krein, K o l m o g o r o v ) Let m be a finite Borel m e a s u r e on 0,1 and m a the a b s o l u t e l y c o n t i n u o u s part w i t h respect to the L e b e s g u e measure. a N o N 1 1 + ~ . a e2~ikt~ 2 k=l k m(dt) = exp(J l o g ~ a dt ) o N o w we are in the p o s i t i o n to a p p l y this result to our f o r g o i n g theory. Suppose (Xn) n 6 Z is a s t a t i o n a r y sequence of r a n d o m v a r i a b l e s d e f i n e d en a p r o b a b i l i t y space (F,B,p). Let m be the c o r r e s p o n d i n g spectral measure.

And corollary 6 . 1 . In the preceding section we have developed another criterion for a stationary sequence to be completely indetermined. 3. Let g(t)~_ 0 be a B-integrable g(t) ) 0 on a set of positive B-measure. function and There exists h(t) such that g(t) = I h(t) I 2 and 1 h(t) = ~ c e 2vikt iff ~ log g(t) dt is finite. ) is the corresponding PREDICTORS g(t) dt spectral measure. 1. SEQUENCES spectral measure m. ,and that g(t) = I h(t) ~ 2 - 3? ,~lkl Vk+n 9 We have 1 I(V o) = _ _ , Vo e~o emlZikt (13) l(V k) = e2Wiktl(Vo ) = Obviously ' VkE ~ k (xe z) -n k=--@@~l k| Vk+n e ~o (n -~ I) and Xn-- ~ ~ikiVk+n = ~ ~iklVk+n A-3~o k=- =0 k=-n+ 1 sg that we have split up Xnintotwo parts, one of them b ~ i n g an element of ~o while the other is orthogonal to ~ o .

3 presently exists no solution. If G o = ( . . r - l , r ) say that an'Ordinary Prediction Problem" solution is given. The is known if the sequence is completely we have developed ,then we it in ~ 8. Now, finally, inaetermined: if ~ # 3cc is a finite set, then we are given an "Interpolation Problem". In what follows we are going to give an answer for the latter problem. For the rest of the section let us suppose isanonempty Remark: finite set. I. If the sequence X~(~) and that s is a fixed integer.

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