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] 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-X|>2-k]<2-k for m, n ≥n. Analyze P[IX-Xn₂>2-k]. (b) Show that X,₁ →p X.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.7: Introduction To Coding Theory (optional)
Problem 6E: Suppose the probability of erroneously transmitting a single digit is P=0.03. Compute the...
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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[X-Xl>2-k].
→P
X.
(b) Show that Xn
Transcribed Image Text: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[X-Xl>2-k]. →P X. (b) Show that Xn
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