Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions Page: 19
View a full description of this dissertation.
Extracted Text
The following text was automatically extracted from the image on this page using optical character recognition software:
The notation p(X, Y), for random variables X and Y, denotes the Prohorov distance
p((X), (Y)) between (X) and (Y), the laws of X and Y, respectively.
Lemma 2.22 places a bound on the Prohorov distance between a probability mea-
sure p, with m = f x dp(x), and the Dirac measure 6,.
Lemma 2.22 (Bloomer, [5]) Suppose X is a random variable with m = E(X) and
V = Var(X). Then,
p(X 5M) < -V. (2.44)
Using Lemma 2.22 and the following notion of the balayage random variable, a
bound on the Prohorov distance between a random variable and its nth level SBA
approximation is obtained in Theorem 2.28.
Definition 2.23 (Hill and Kertz, [18]) Let Y be an integrable random variable and
let a,b be constants such that -oo < a < b < oo. The (balayage) random variable, Yab,
is equal to Y if Y V [a, b], is equal to a with probability (b- a)-1 fYE[a,b] (b- Y)dP,
and is equal to b with probability (b- a) -1 fYE[a,b] (Y a)dP.
Note that (b - a)-1 fYe[a,b] (b - Y) dP + (b - a)-1 'fY[a,b] (Y - a) dP = P(Y E [a, b]).
Hill and Kertz [18] note that Yab is the random variable with maximum variance
which coincides with Y off [a, b], and which has expectation E(Y). This statement
can be extended to the following: If Y takes only positive values, Yab is the random
variable with maximum kth moment, k > 1, which coincides with Y off [a, b], and
which has expectation E(Y). Denote the kth moment of Yab by E(Yab k
Lemma 2.24 Let Y be a nonnegative integrable random variable and 0 < a < b < 00.
Then E(Yk) < E(Yab ), for k > 1, with equality when k= 1.
The proof follows directly from the proof given by Hill and Kertz [18] (Lemma 2.2).19
Upcoming Pages
Here’s what’s next.
Search Inside
This dissertation can be searched. Note: Results may vary based on the legibility of text within the document.
Tools / Downloads
Get a copy of this page or view the extracted text.
Citing and Sharing
Basic information for referencing this web page. We also provide extended guidance on usage rights, references, copying or embedding.
Reference the current page of this Dissertation.
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/m1/26/: accessed April 26, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .