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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 $ ?

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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

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