Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions Page: 3
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Chapter 2
Background
2.1 The Sequential Barycenter Array Construc-
tion
The sequential barycenter array (SBA) construction gives a general and natural
method for randomly generating probability measures with a prescribed mean. Before
describing the construction, some definitions are in order.
Throughout this work, let X be a real-valued random variable with distribution
function F, such that E[IXI] <cc.
Definition 2.1 The F-barycenter of the interval (a, c], bF(a, c], is given by
f JI (,c]fac] xdF(x) z ~)>Fa
bF(a,c] E[XXE(a,] F(c)-F(a) , if F(c)>F(a)
a if (c)= F(a).
That is, the F-barycenter of (a, c] is the conditional expectation of X over the in-
terval (a, c]. The following lemma characterizes some elementary properties of F-
barycenters.
Lemma 2.2 (Hill and Monticino [19]) Fix a < c such that P[X c (a, c]] > 0, and let
b = bF(a, c]. Then3
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Valdes, LeRoy I. Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions, dissertation, December 2002; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc3304/m1/10/: accessed April 23, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .