Theorem 1.6 (The Kolmogorov inequality) Let X1, X2, Xn be independent random variables with mean 0 and suppose that Var Xk<∞ for all k. Then, for x > 0, P(max Sk>x) ≤ Isk≤n Σ-Var X In particular, if X1, X2,..., X, are identically distributed, then P(max Sx) ≤ Isk≤n nVar X₁ x2

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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Theorem 1.6 (The Kolmogorov inequality)
Let X1, X2,
Xn be independent random variables with mean 0 and suppose
that Var Xk<∞ for all k. Then, for x > 0,
P(max Sk>x) ≤
Isk≤n
Σ-Var X
In particular, if X1, X2,..., X, are identically distributed, then
P(max Sx) ≤
Isk≤n
nVar X₁
x2
Transcribed Image Text:Theorem 1.6 (The Kolmogorov inequality) Let X1, X2, Xn be independent random variables with mean 0 and suppose that Var Xk<∞ for all k. Then, for x > 0, P(max Sk>x) ≤ Isk≤n Σ-Var X In particular, if X1, X2,..., X, are identically distributed, then P(max Sx) ≤ Isk≤n nVar X₁ x2
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