This paper studies the time evolution of the covariance of the position estimates in single-robot Simultaneous Localization And Mapping (SLAM). A closed-form expression is derived, that establishes a functional relation between the noise parameters of the robot's proprioceptive and exteroceptive sensors, the number of features being mapped, and the attainable accuracy of SLAM. Furthermore, it is demonstrated how prior information about the spatial density of landmarks can be utilized in order to compute a tight upper bound on the expected covariance of the positioning errors. The derived closed-form expressions enable the prediction of SLAM positioning performance, without resorting to extensive simulations, and thus offer an analytical tool for determining the sensor characteristics required to achieve a desired degree of accuracy. Simulation experiments are conducted, that corroborate the presented theoretical analysis.