In this paper, we investigate the correlation properties of the squared envelope of a class of autocorrelation-ergodic (AE) sum-of-cisoids (SOC) simulation models for mobile Rayleigh fading channels. Novel closed-form expressions are presented for both the ensemble and the time-averaged autocorrelation functions (ACFs) of the simulation model's squared envelope. Basing on those expressions, we show that under certain conditions, the squared envelope of the SOC model is itself an AE random process. In addition, we evaluate the performance of three fundamental methods for the computation of the model's parameters---namely the generalized method of equal areas (GMEA), the $L_p$-norm method (LPNM), and the Riemann sum method (RSM)---regarding their accuracy for emulating the squared envelope ACF of the channel. The obtained results can be used as a basis to design efficient simulators for the performance analysis of mobile communication systems sensitive to the correlation properties of the channel's squared envelope.