In this letter, the closed form sum rate for the random beamforming is analyzed over the downlink multiple-input single-output and multiple input multiple-output interfering broadcast channel with finite users. The analysis approach is based on the limiting throughput distribution theorem, and can be applied for the multi-cell interference scenarios with heterogeneous signal to interference-plus-noise ratio (SINR) distributions and user schedule probabilities. It is proved that both the exact and the asymptotic sum rates achieved from SINR/signal-to-interference ratio analysis will converge to the Gumbel type distribution. The effectiveness of the proposed analytical framework is confirmed by simulation results.