We thoroughly investigate the asymptotic distribution of sum of multiple signal-to-interference-plus-noise-ratios (SINRs) that could be applied to the outage performance evaluation of integrated non-orthogonal multiple access (NOMA) and hybrid automatic repeat request (HARQ) systems. It is proved that the asymptotic distribution of sum of SINRs is a piecewise function of the transmission rate and power allocation coefficient. By leveraging the integral transform, convolution theorem, and Dirac delta function, the asymptotic distribution of sum of SINRs can be recursively calculated. Moreover, the philosophy of Yang Hui's triangle is employed to derive the asymptotic distribution in closed-form. Monte Carlo simulations are finally performed to verify our analysis.