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I wonder if the following are equivalent.

1) E(|X|) < infinity

2) I(|X| > n) goes to 0 as n goes to infinity (I is the indicator function)

3) P(|X| > n) goes to 0 as n goes to infinity.

1) => 2) and 2) => 3) are easy to see, please help me to show 2) => 1) and 3) => 2) if they are indeed true. Thanks.

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# Convergence of indicator functions for L1 r.v.

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