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This paper studies the achievable performance of a single-antenna spectrum-sharing system. Due to limited cooperation between secondary users (SUs) and primary users (PUs), perfect channel state information (CSI) related to the PUs is hard to obtain for the SUs. Therefore, we assume the secondary transmitter and receiver have only statistical CSI from the secondary transmitter to the primary receiver. First, we investigate the optimal power adaptation method under both a peak transmit power constraint and an average interference constraint. The optimal power allocation strategy is categorized into three regions based on the channel variations. Furthermore, we propose an efficient method in order to determine the optimal thresholds to divide three regions. Second, we derive closed-form expressions of the ergodic capacity of the secondary link over Rayleigh fading channels. The analytical results reveal some interesting insights into the achievable performance of the spectrum-sharing system, which are discussed and verified by simulation results.