Random distribution functions

Abstract

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