A Century of Mathematics in America (History of Mathematics, by Peter Duren

By Peter Duren

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Extra info for A Century of Mathematics in America (History of Mathematics, Vol 3)

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This Markov probability is constructed in the usual way, by specifying the fonoJard transition probabilities and the initial distribution. 3) and lifYn=Y~ o otherwise lve claim that this system (mP n' m,; n) is a system of forward tran- sition probabilities. 2) since and all terms in the sum must be equal to zero. Clearly mPn(Ym'Y n ) ~O and if Pm(y m) =O,l: mPn(Ym'Y n ) = l:Pn(y n ) =1. 2).

Further we shall assume that the families we consider satisfy the property that This assumption is not a severe restriction. We shall think of the spaces (~n) n € IN and the way they are related to each other (the projections :Emn) as a "repetitive structure" and give a precise definition later. 29 2. Notions of sufficiency The usual notion of sufficiency is defined for one single sample space X as in chapter I and does not take into account the presence of a repetitive structure. It is therefore as such not adequate (detailed enough) to make our further investigations fruitful.

The only way in which P comes into the picture is via the maps Qt ' and they do not detern mine the family P in general (they do if we demand P to be maximal or extremal, see next section). The maps mQn determine the con- ditional distribution of Yn given Ym, see section 3. A similar result (weaker) can be derived for the more "direct" t diagrams involving II n instead of Qt . 5 Proposition I f ! 4. 4) that for ~n PEP n , t t £ t t £ Q n (P ) = n mn mnQ n n (P ) = n mn rnn (P ), m n ~n tn ~n ~n which was to be proved.

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