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In this paper, the downlink performance of distributed antenna systems (DAS) with random antenna layout is investigated. We consider the composite channel including large-scale fading and small-scale fading. When the large-scale channel state information, which usually varies slowly and is easy to be obtained, is available at the transmitter, the problem of power allocation among distributed antennas with the target of downlink capacity maximization is formulated. Based on system scale-up, we derive a precise approximation of the downlink ergodic capacity by adopting random matrix theory. We also propose an iterative method to calculate the unknown parameter in the approximation. Moreover, the approximation is proved to be concave on the transmit powers of the distributed antennas. Consequently, a simple sub-optimal power allocation scheme is proposed, with which the system capacity is illustrated to be quite close to the optimal one obtained by numerical optimizations.