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In this paper, we propose a distributed resource allocation framework for cognitive radio networks by using the orthogonal frequency division multiple access (OFDMA) modulation. We jointly consider the constraints of quality of service (QoS), maximum power, and minimum rates in the network to obtain optimal available subcarrier sets and transmission power. The fairness of resource allocation is guaranteed by incorporating the probability that a subcarrier is occupied into the link capacity expression. We present a distributed subcarrier selection and power allocation algorithm and evaluate the algorithm through simulations. Our results confirm that the proposed algorithm outperforms the existing algorithms in terms of throughput, the number of secondary links admitted, and the fairness of resource allocation.