Convergence of indicator functions for L1 r.v.

[jk_zhengli]

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.

 

[micromass]

Does this hold for the Cauchy distribution? http://en.wikipedia.org/wiki/Cauchy_distribution