On geometric recurrence for time-inhomogeneous autoregression

The time-inhomogeneous autoregressive model AR(1) is studied, which is the pro-cess of the form Xn+1 = & alpha;nXn + & epsilon;n, where & alpha;n are constants, and & epsilon;n are independent random variables. Conditions on & alpha;n and distributions of & epsilon;n are established that guarantee the geomet-ric recurrence of the process. This result is applied to estimate the stability of n-steps tran-sition probabilities for two autoregressive processes X(1) and X(2) assuming that both & alpha;(i) n , i & ISIN; {1, 2}, and distributions of & epsilon;(i) n , i & ISIN; {1, 2}, are close enough.