This paper explores receiver autonomous integrity monitoring (RAIM) for sequences of filtered measurements. Optimal position estimation for navigation systems with known dynamics (such as carrier phase positioning or integrated GPS/INS navigation) is provided by filtering measurements over time. All measurements within the time-sequence are vulnerable to faults. There is currently no widely implemented algorithm for the detection of faults that last over multiple epochs. In this work, two sequential residual-based RAIM algorithms are investigated. The first algorithm is a batch-type procedure. The batch least-squares residual is computed at each epoch using a sliding-window mechanism. It is derived in a compact formulation using a forward-backward smoother (FBS). The iterative FBS residual generation process includes a residual norm weighting procedure accounting for measurement error correlation and prior knowledge of state variables. The second method, based on a Kalman filter (KF), is truly sequential. A KF detection test statistic is defined and its probability distribution is established (assuming time-uncorrelated normally distributed measurement noise). The KF RAIM residual is carefully derived to enable probability of missed-detection determination at each time step. In addition, in contrast to the KF implementation, the sampling interval within the batch may be treated as an extra navigation design parameter: increasing the batch sampling interval decreases the estimation performance but lowers the batch computation load. In this research, the fault-detection performance sensitivity to sampling interval is investigated. The analysis yields counter-intuitive results in the presence of time-correlated measurement errors. Finally, both batch and KF residual-based RAIM methods are evaluated for a benchmark application of aircraft precision approach, where differential GPS and Galileo code and carrier phase measurements are filtered for floating cycle ambiguity estimation.