# Convergence in series of expectation

If $X_i$ are independent and the series of $\sum X_i$ is convergent. $\sum X_i = Y$ , does it imply $\sum EX_i = EY$ ?

-
What do you think? – Did Nov 3 '12 at 21:13
I think no, because it does not satisfy the kolomogorov test, for example I think a series of Cauchy distributions can be made such that they tend to zero ( fast enough ) such that they have a good convergence in series to zero but the expectation of the sum becomes zero ( where the sum of expectations are undefined) but one of the 3 conditions of kolpmogrov is hard to check for this distribution! I need some other non-negative symmetric distribution which has infinite expectation but it is for example discrete such that I can deal with it easier! – peanut Nov 3 '12 at 21:17
Cauchy distributions do not have an expectation, and nor do their sum (which is another Cauchy distribution) – Henry Nov 3 '12 at 21:33
For Kolomogorov test, only expectation of $\sum X_i1_{X_i <A}$ should be convergent which is zero. – peanut Nov 3 '12 at 21:35
You might want to elaborate the question slightly so it is not confused with a more elementary question about linearity of expectation... – copper.hat Nov 3 '12 at 22:07