Advances in communications technologies entail demands for higher data rates. One of the popular solutions to fulfill this requirement was to allocate additional bandwidth, which unfortunately is not anymore viable due to spectrum scarcity. In addition, spectrum measurements around the globe have revealed the fact that the available spectrum is under-utilized. One of the most remarkable solutions to cope with the under-utilization of radio-frequency (RF) spectrum is the concept of cognitive radio (CR) with spectrum sharing features, also referred to as spectrum sharing systems. In CR systems, the main implementation issue is spectrum sensing because of the uncertainties in propagation channel, hidden primary user (PU) problem, sensing duration and security issues. Hence, the accuracy and reliability of the spectrum sensing information may inherently be suspicious and questionable. Due to the imprecise spectrum sensing information, this dissertation investigates the performance of an orthogonal frequency-division multiplexing (OFDM)-based CR spectrum sharing communication system that assumes random allocation and absence of the PU's channel occupation information, i.e., no spectrum sensing is employed to acquire information about the availability of unused subcarriers or the PU's activity. In addition, no cooperation occurs between the transmitters of the PUs and secondary users (SUs). The main benefit of random subcarrier utilization is to uniformly distribute the amount of SUs' interference among the PUs' subcarriers, which can be termed as interference spreading. The analysis and performance of such a communication set-up provides useful insights and can be utilized as a valid benchmark for performance comparison studies in CR spectrum sharing systems that assume the availability of spectrum sensing information. In the first part this dissertation, due to the lack of information about PUs' activities, the SU randomly allocates the subcarriers of the primary network and collide with the PUs' subcarriers with a certain probability. 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. In the presence of multiple SUs, the multiuser diversity gain of SUs is also investigated. To avoid the subcarrier collisions at the SUs due to the random allocation scheme and to 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. In the second part of this dissertation, in addition to the collisions between the SUs and PUs, the inter-cell collisions among the subcarriers of SUs (belonging to different cells) are assumed to occur due to the inherent nature of random access scheme. A stochastic analysis of the number of subcarrier collisions between the SUs' and PU's subcarriers assuming fixed and random number of subcarriers requirements for each user is conducted. The performance of the random scheme in terms of capacity and capacity (rate) loss caused by the subcarrier collisions is investigated by assuming an interference power constraint at PU to protect its operation. Lastly, a theoretical channel fading model, termed hyper fading channel model, that is suitable to the dynamic nature of CR channel is proposed and analyzed. To perform a general analysis, the achievable average capacity of CR spectrum sharing systems over the proposed dynamic fading environments is studied.