Analysis Of Sequential Barycenter Random Probability Measures via Discrete Constructions Page: 34
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2n-1 2
[n] 2 i2 n,2i
+3 pi7,2i-2 Mm.,2i-1 1)+m2 KTm,2i-1 -n,2i-22
n,2i-1 2 n,2i
E n,2i-2 E~n,2i - En,2i-2,2-1 < 6 . (2.84)
Then
max C3(n)P1o> )Ao (y) < maX 3n)o, (y + 6)+ 1 - RSn,3 ()
60max ~ [1ItO~ 3 (6)J f <U3 K) a(jU1Y3)+
(2.85)
Proof: Let 6 > 0 and let SD = E(X()3) - E(X3) + 31E(X(n)2) - E(X2). Recall
that E(X) - E(X(n)) and E(X) < 1. ThenBo, I{X :E(X - E(X))3 < y}
< BIo,{0{X E(X(n) -E(X()))3
K Bo0, {X E(X () -E(X(n))3
Bo0I{X E(X(n) - E(X(n))3
< Bo0,1{X E(X(n) -E(X(n)))3
< Co0'(y + 6) + 1 f- RSof (6).< y + E(X () - E(X(")))3 - E(X - E(X))31}
< y + SDn(X), SD"(X) < 6} +
< y + SD!](X), SDn(X) > 6}
< y + 6} + Bo,, {X SDK(X) > 6}
(2.86)Inequality (2.86) follows from Lemma 2.37. Also, for 6 > 0,
B1o, I{X: E(X- E(X)) 3 < Y}
{> B4X : E(X(") -E(X ")))3<
> Bo1{X: E(X(n) -E(X ")))3< y
> B 4{ X : E (X (n) - E(x(X))) 3
Bo,1{X : SDK(X) <6} 1_1
> C0'(y 0 - 6)- [1 - RS""()].SD"(X)}
SD (X), SD"(X) < 6}
6}+(2.87)
Inequality (2.87) follows from the Bonferroni inequality. Since inequalities (2.86) and
(2.87) each holds for all 5 > 0, then (2.85) holds true. O:
Corollary 2.41 places bounds on the fourth central moment.34
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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/m1/41/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .