Recent analysis of linear multiuser receivers for DS-CDMA signals has considered signal-to-interference (SIR) performance in large systems with each user employing random spreading sequences. Unfortunately, it is only in the special case of the received powers of all users being equal that a resulting integral equation for SIR has an explicit solution with demonstrable approximation accuracy for finite-sized systems. This paper presents an alternative approach to calculating the SIR performance of MMSE receivers which relies primarily on the law of large numbers, and provides an estimate for SIR that is valid for arbitrary distributions of received interferer powers.