Markov-chain Monte--Carlo methods have been demonstrated to offer an attractive alternative to the design of approximate (near optimal) maximum a-posteriori (MAP) detectors for synchronous direct-sequence code-division multiple access (DS-CDMA) and multi-input, multi-output (MIMO) multiple antenna applications. Central to evaluating these method is understanding their convergence properties. In other works, this has been established via simulation, and the underlying theoretical basis has been identified. The contribution of this paper is to extend the theoretical understanding by rigorously establishing both convergence and convergence rate results for a wide class of Metropolis--Hastings methods.