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This work makes the preliminary contribution of studying the asymptotic performance of a multi-user intelligent reflecting surface (IRS) assisted-multiple-input single-output (MISO) downlink system under imperfect CSI. We first extend the existing least squares (LS) ON/OFF channel estimation protocol to a multi-user system, where we derive minimum mean squared error (MMSE) estimates of all IRS-assisted channels over multiple sub-phases. We also consider a low-complexity direct estimation (DE) scheme, where the BS obtains the MMSE estimate of the overall channel in a single sub-phase. Under both protocols, the BS implements maximum ratio transmission (MRT) precoding while the IRS design is studied in the large system limit, where we derive deterministic equivalents of the signal-to-interference-plus-noise ratio (SINR) and the sum-rate. The derived asymptotic expressions, which depend only on channel statistics, reveal that under Rayleigh fading IRS-to-users channels, the IRS phase-shift values do not play a significant role in improving the sum-rate but the IRS still provides an array gain. Simulation results confirm the accuracy of the derived deterministic equivalents and show that under Rayleigh fading, the IRS gains are more significant in noise-limited scenarios. We also conclude that the DE of the overall channel yields better performance when considering large systems.