Properties of binary autoregressive sequences

It is shown that 1) a binary autoregressive sequence generated by yn= xn⊕ yn-N, where N = 2^Kand K = 0, 1,..., is wide-sense Markov and 2) under the condition Pr [xn= 1] = p < 0.5, the error associated with a one-step prediction of a first-order Markov sequence yn= Xn⊕ yn-1grows with the order N of the predictor. The error attains its minimum value when N = 1.