Convergence of indicator functions for L1 r.v.

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The discussion centers on the equivalence of three conditions related to the convergence of indicator functions for L1 random variables. It is established that E(|X|) < infinity implies that I(|X| > n) approaches 0 as n increases, and that I(|X| > n) going to 0 leads to P(|X| > n) going to 0. The main inquiry is whether the reverse implications hold, specifically if I(|X| > n) going to 0 guarantees E(|X|) < infinity, and if P(|X| > n) going to 0 ensures I(|X| > n) goes to 0. Additionally, the discussion raises a question regarding the validity of these implications for the Cauchy distribution. The thread seeks clarification on these mathematical relationships.
jk_zhengli
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Hi all,

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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