If for some en 0 with e n yen , it holds that P( X n 1 X n sup3 e n ) yen , then show that X n to a r v It suffices to show that X n converges mutually a s To this end, set A n ( X n 1 X n sup3 e n ), n sup3 1, and use Exercise 3 in this chapter in order to conclude that 0 Then, by setting Auml A n and N c Auml , it follows that the event N c (with P( N c ) 1) is the set over which X n converges mutually n 1

Question: If for some en > 0 with e n < ¥, it holds that P(| X n+1 X n | ³ e n ) <

If for some en > 0 with

e_n< ¥, it holds that

P(|X_n+1€“X_n| ³ e_n) < ¥, then show thatX_n

to a r.v.

It suffices to show that {X_n} converges mutually a.s. To this end, set A_n = (|X_n+1 €“ X_n| ³ e_n), n ³ 1, and use Exercise 3 in this chapter in order to conclude that = 0. Then, by setting Ä€ = A_n and N^c = Ä€, it follows that the event N^c (with P(N^c) = 1) is the set over which {X_n} converges mutually.

n=1

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