### Constructions of random distributions via sequential barycenters

Theodore Hill, Michael Monticino
1998 The Annals of Statistics
This article introduces and develops a constructive method for generating random probability measures with a prescribed mean or distribution of the means. The method involves sequentially generating an array of barycenters which uniquely defines a probability measure. Basic properties of the generated measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almost all have finite support. Applications are
more » ... to models for average-optimal control problems and to experimental approximation of universal constants. Introduction. The purpose of this note is to introduce a general and natural method for constructing random probability measures with any prescribed mean or distribution of the means. This method complements classical and recent constructions [e.g., Dubins and Freedman (1967), Ferguson (1973, 1974) , Graf, Mauldin and Williams (1986), Mauldin, Sudderth and Williams (1992) and Monticino (1996)], none of which generates random measures with a priori specified means. In fact, even the calculation of the distribution of the means for those constructions is difficult [cf. Cifarelli and Regazzani (1990) and Monticino (1995)]. The new method presented here, which is based on sequential barycenters, satisfies Ferguson's (1974) two basic requirements that such constructions have large support and be analytically manageable. The construction is easy to implement and is robust, allowing generation of random measures which are either (almost surely) discrete or continuous, as desired. Since many problems in probability and analysis involve distributions with given means, the new construction will perhaps prove a useful tool in a variety of applications. Sequential Barycenter Arrays. This section introduces the notion of a sequential barycenter array (SBA) and develops some basic properties of the probability measures defined by the arrays. These SBA's, although not named as such, are used in standard proofs of Skorohod's embedding theorems [e.g., Billingsley (1986), Section 37], and it is the reversal of this standard procedure which is the foundation for the construction of the random measures given in the next section. This article introduces and develops a constructive method for generating random probability measures with a prescribed mean or distribution of the means. The method involves sequentially generating an array of barycenters which uniquely defines a probability measure. Basic properties of the generated measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almost all have finite support. Applications are given to models for average-optimal control problems and to experimental approximation of universal constants. Throughout this section, let X be a real-valued random variable with distribution function F, such that E[IXI ] < oo. DEFINITION 2.1. The F-barycenter of (a, c], bF(a, c], is given by f Fa, cif(,c]xdF(x) (,E[XIX E (a, c]] = (a c] x dF() if F(c) > F(a), a, if F(c) = F(a). Some elementary properties of F-barycenters are recorded in the next lemma. LEMMA 2.2. Fix a < c such that P[X E [a, c]] > 0 and let b = bF(a, c]. Then: (i) F(c) > F(a) if and only if b > a; (ii) (F(c) -F(a))b = (F(b) -F(a))bF(a, b] + (F(c) -F(b))bF(b, c]; (iii) bF(a, b] = b if and only if bF(b, c] = b; (iv) b > bF(a, x], for all x E (a, c]. DEFINITION 2.3. The sequential barycenter array (SBA) of F is the triangular array {mn, k}n=l k-l = {mn, k(F)} = M(F) defined inductively by (2.1) ml, 1 = E[X] = xdF(x) = bF(-oo, oo), (2.2) mn,2j = mn,j, for n > 1 and j = 1,..., 2n-1-1, (2.3) mn,2j-1 = bF(mn-1, j-1, n-, j], for j = 1,...,2"-1 with the convention that mn, = -oo and mn, 2n = 00. EXAMPLE 2.4. Suppose X is uniformly distributed over [0, 1]. Then {Mn,k(F)} =