The rate distortion functionR(D)is calculated for two time-discrete autoregressive sources--the time-discrete Gaussian autoregressive source with a mean-square-error fidelity criterion and the binary-symmetric first-order Markov source with an average probability-of-error per bit fidelity criterion. In both cases it is shown thatR(D)is bounded below by the rate distortion function of the independent-letter identically distributed sequence that generates the autoregressive source. This lower bound is shown to hold with equality for a nonzero region of small average distortion. The positive coding theorem is proved for the possibly nonstationary Gaussian autoregressive source with a constraint on the parameters. Finally, it is shown that the rate distortion function of any time-discrete autoregressive source with a difference distortion measure can be bounded below by the rate distortion function of the independent-letter identically distributed generating sequence with the same distortion measure.