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Unformatted text preview: 4. Let { A n } be a sequence of events with lim n P [ A n ] = 0 and ∑ ∞ n =1 P [ A n \ A n +1 ] < ∞ . Show that P [ A n i.o.] = 0. [Hint: ∪ m k = n A k = ∪ m1 k = n [ A k \ A k +1 ] ∪ A m .] 5. In this problem X is a nonnegative random variable. (a) Suppose there are constants p > 0 and C > 0 such that P [ X > n ] ≤ C · np ,n = 1 , 2 ,.... Show that E [ X r ] < ∞ provided 0 < r < p . (b) Suppose that r > 0 and that E [ X r ] < ∞ . Show that lim x →∞ x r P [ X > x ] = 0 ....
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This note was uploaded on 12/11/2011 for the course MATH 280a taught by Professor Driver,b during the Fall '08 term at UCSD.
 Fall '08
 Driver,B
 Math, Probability

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