Asymptotically homogeneous iterated random functions with applications to the HARCH process (Q1041395)

Iterative time series on a finite dimensional space are considered of the form \(X_n=F(X_{n-1},\varepsilon_n)\), \(X_0=x\), where \(\varepsilon_i\) are i.i.d., \(F(\cdot,w)\) is an asymptotically homogeneous function, i.e. \(F(t_n x_n,w)/t_n\to G(x,w)\) as \(t_n\to\infty\), \(x_n\to x\), \(G\) is some function. The authors establish conditions under which the Markov chain \(X_n\) is positive Harris recurrent. The key condition is \[ \lim_{m\to\infty}{1\over m}\sum_{i=1}^m E\ln\|G(Z_{i-1},\varepsilon_i)\|<0, \] where the auxiliary chain \(Z_i\) is defined by \(Z_i=G(Z_{i-1},\varepsilon_i)/\|G(Z_{i-1},\varepsilon_i)\|\). This result is applied to derive conditions of stationarity of HARCH(1) process.