A general three-series theorem

Abstract:

Let $\{ \Omega ,\mathcal {F},P\}$ be a probability space. The subset of $\Omega$ on which an arbitrary sequence of random variables converges is shown to be equivalent to the intersection of three other sets, each specified by the almost sure convergence of a certain sequence of random variables. Kolmogorov’s three-series theorem, which gives necessary and sufficient conditions for the almost sure convergence of a sequence of sums of independent random variables, is obtainable as a particular case of the present result.