The heavy-tailed nature in dynamic spectrum access networks challenges the applicability of conventional network stability criterion. To counter this problem, a new stability criterion, namely moment stability, is introduced, which requires that all secondary users (SUs) with light-tailed traffic have bounded queueing delay with finite mean and variance. This stability criterion can prevent heavy-tailed traffic, e.g., video and Internet traffic, from degrading the QoS performance of light-tailed ones, e.g., email deliveries and audio/voice traffic. The critical conditions for the existence of a scheduling policy to achieve moment stability are derived. The network stability region yielded from these conditions is shown to be directly related to the primary and secondary user activities, the number of SUs, and the total number of primary user channels available to SUs. Moreover, it is shown that the maximum-weight-alpha scheduling algorithm, which makes the scheduling decision based on the queue lengths raised to the alpha-th power, is throughput optimal with respect to moment stability. It is proven that its throughput optimality holds independent of the stochastic properties of SU traffic including its marginal distribution (i.e., heavy-tailed or light-tailed distribution) and its time correlation structure (i.e., long range or short range dependence).