Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions Page: 12

                                
which is equal to F.

This example illustrates the general case which will be shown in Theorem 2.20: If X
has support on a set of k points, then X(-) - X, for all n > k. Note that due to
the binary nature of the SBA approximation, it might seem that X(") should equal
X for any n that 2n > k. However, Example 2.14 shows that it may be necessary to
have n > k before X(=) X.
A few preliminary definitions are needed before proving Theorem 2.20. Denote by
B(R) the Borel subsets of R. Suppose B e B(R) and P(B) $ 0, then the conditional
random variable XB of X is the random variable such that
P[XBcA]  P[X   cAnB]
P[x Xs  A]=
P[X e B]
for A c B(R). Denote the cdf of XB by FB.
The notion of a nth level SBA decomposition of the measure of a random variable
is given next.
Definition 2.15 Let X be a random variable with cdf F, distribution p, and SBA
{r~k}  2-1 For n > 1 and 1 < i < 2~-1 let
{Mn~k n=1 k=l"     --,
X~ j(2.6)
X    X(mn,2i-2,mn,2i                    (2.6)
and denote the distribution of X}n by [4n. The nth-level SBA decomposition of p is
given by
2n-1
p n] n].                        (2.7)
i=-1
Note that, for A e B(R),
(2n-1             2 n-1
p(A)       Pn]   (A)      p    "P(A).          (2.8)
)i=1             i= 1
Using the inversion formula given in Theorem 2.8, the values {pI } can be calcu-
lated from the SBA of X without explicitly knowing p. Also, note that pi" has density

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