If for some en > 0 with e n < ¥, it holds that P(| X n+1
Question:
en< ¥, it holds that
P(|Xn+1€“Xn| ³ en) < ¥, then show thatXn 
to a r.v.It suffices to show that {Xn} converges mutually a.s. To this end, set An = (|Xn+1 €“ Xn| ³ en), n ³ 1, and use Exercise 3 in this chapter in order to conclude that 
= 0. Then, by setting Ā = 
An and Nc = Ā, it follows that the event Nc (with P(Nc) = 1) is the set over which {Xn} converges mutually.
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It suffices to show that X n converges mutually as That is for every 0 and every N c with P N 0 ther...View the full answer
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Related Book For 
An Introduction to Measure Theoretic Probability
ISBN: 978-0128000427
2nd edition
Authors: George G. Roussas
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