Two ergodicity criteria for stochastically recursive sequences (Q1323523)

For a stochastically recursive sequence \(X(n+1) = f(X(n), Z(n))\), where \((Z(n))\) is stationary and metrically transitive, criteria for coupling and strong coupling convergence to a stationary sequence are proved. The fundamental property for these theorems to hold is: An event \(A \in \sigma (Z(k) : k = m + n)\) is renovating on the time interval \([n,n + m]\) for \((X(n))\) if \(X(n + m + 1) = g(Z(n), \dots, Z(n + m))\) for some function \(g\) on the set \(A\). The proofs depend on the notion of renovating events and renovating sequences of events, and on the fact that the existence of a stationary sequence of renovating events for \((X(n))\) (with the same \(g)\) is equivalent to strong coupling convergence.