On geometric recurrence for time-inhomogeneous autoregression

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