Random distribution functions

摘要: How cani oine choose, at ranidom, a probability measture oni the Ullit interval? This paper develops answer anniiounced in [4]. Section 1, w-hich can be skipped without logical loss, gives fairly full but slightly iniformal accounit. The formalities begini with section 2. All later sections are largely inidepelndent of one another. Sectioin 10 iiidexes definiitions made sectioni used other sections. A distribution function F on closed uniit initerxal I is nonidecreasing, nonniiegative, real-valued funcetioni onl I, normalized to 1 and coIntinuous from right. To each t.here corresponds anid onily onie measure IFl Borel subsets F(x) e(qual IFI-measure interval [0, x], for all x G I. Choosing ranidom therefore tantamounlt choosing random distributioni fulnctioll measurable map space (Q, i, Q) funictions unit where endowed its natural a-field, that is, a-field genierated by customary weak* topology. F, namely QF-', prior A. Of course, essentially stochastic process 'Ft, 0 < t I" Q), whose sample funcetioiis funictionis: F,(w) F(w) evaluated at, t. T'herefore, t'his thought as dealinig witlh class funietionis, processes, or probabilities. Similar priors treated in. [9], [11], [16], [17]. Sinice indefiniite initegral funetioni convex, this also convex funcetioiis, we do nlot pursue idea. Which funietionis does study? base j. ont unlit s(lqlare S, assigling cornlers (0, 0) (1, 1). Eachi suchl ,u determinies distributioii so P), A, which will niow described, explaininig how select xvalue distributionl funictioII random. ASSUMPTION. For ease exposition, assume throughout IA concentrates on, assigns to, interior S.