The 1-stable random variables have the Cauchy distribution.  So it is easy to show by an explicit calculation that they are in weak L 1, because if X is 1-stable, then X^(t) = A cot(pi t/2) for some constant A.

Now if X1, X2, \dots Xn are identically distributed independent Cauchy random variables, then sum ai Xi has the same distribution as A X, where A = sum |ai|.

Thanks. I finally figured this out. Good to hear from you. I hope all is well in MO.