On geometric recurrence for time-inhomogeneous autoregression

The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form Xn+1=αnXn+εn, where αn are constants, and εn are independent random variables. Conditions on αn and distributions of εn are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes X(1) and X(2) assuming that both αn(i), i∈{1,2}, and distributions of εn(i), i∈{1,2}, are close enough.