Answered: 20.21. Suppose that the sequence (X) is… | bartleby

Algebra & Trigonometry with Analytic Geometry

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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20.21. Suppose that the sequence (X) is fundamental in probability in the sense that
for e positive there exists an N such that P[IX-X|>e] <e for m, n > N.
(a) Prove there is a subsequence (X) and a random variable X such that
lim XX with probability 1. Hint: Choose increasing n such that
P[IX-X₁>2-k]<2-k for m, n ≥n. Analyze P[IX-Xn₂>2-k].
(b) Show that Xn
→P
X.

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