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This paper investigates the performance of an orthogonal frequency-division multiplexing (OFDM)-based cognitive radio (CR) spectrum sharing communication system that assumes random allocation and absence of the primary user's (PU) channel occupation information, i.e., no spectrum sensing is employed to acquire information about the availability of unused subcarriers. In case of a single secondary user (SU) in the secondary network, due to the lack of information of PUs' activities, the SU randomly allocates the subcarriers of the primary network and collide with the PUs' subcarriers with a certain probability. To maintain the quality of service (QoS) requirement of PUs, the interference that SU causes onto PUs is controlled by adjusting SU's transmit power below a predefined threshold, referred to as interference temperature. In this paper, the average capacity of SU with subcarrier collisions is employed as performance measure to investigate the proposed random allocation scheme for both general and Rayleigh channel fading models. Bounds and scaling laws of average capacity with respect to the number of SUs, PUs, and available subcarriers are derived. In addition, in the presence of multiple SUs, the multiuser diversity gain of SUs assuming an opportunistic scheduling is also investigated. To avoid the interference at the SUs that might be caused by the random allocation scheme and obtain the maximum sum rate for SUs based on the available subcarriers, an efficient centralized sequential algorithm based on the opportunistic scheduling and random allocation (utilization) methods is proposed to ensure the orthogonality of assigned subcarriers.