On sums of symmetrically dependent random variables

In a paper [1] I have proved some theorems concerning the signs of sums of symmetrically distributed random variables. Later in a note [2] I have stated some results on the signs of sums of independent random variables. It was promised that complete proofs should be publisbed elsewhere. In this paper the results in [1] are generalized. Furthermore the results proved by D. A. Darling in [4] for independent variables are extended to symmetrically dependent variables. A complete proof of one of the results stated in [2] is also given. This result, however, is restricted to independent variables. It is the idea in the original proof of this result, which now forms the basis of the proof of Theorem 1 below.