Regarding the converse stated in Exercise 6, if the events A n ,n 1, are not
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Regarding the converse stated in Exercise 6, if the events An,n ≥ 1, are not independent, then P(lim supn→∞ An) = 0 need not imply that ∞. Give one or two concrete examples to demonstrate this assertion. Take (Ω, A, P) = ((0, 1), B(0,1), λ),λ being the Lebesgue measure. Then
(a) Take
and show that
so that
Then, by Exercise 5, P(lim supn→∞, An) = 0, where An = (|Xn| ≥ l/k) for any arbitrary but fixed k = 1, 2,... Also, show that
(b) Take
and show that
so that
Again, P(lim sup n→∞ An) = = , as in (a). Also show that
P(An) < ∞.
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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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