Approximations to sums of random variables (RVs) have been a longstanding issue in view of the usual intricacy of the exact sum statistics. In this letter, a new technique called asymptotic matching is introduced and a corresponding new framework is proposed that allows for accurate statistical approximations to a broad class of sums of positive correlated RVs. As an application example, the proposed framework is used to analyze the performance of an equal-gain combiner operating over correlated Rayleigh fading channels, rendering an outstanding fit, especially at high signal-to-noise ratio - a most representative operational regime for communication systems.