On the extremal behavior of sub-sampled solutions of stochastic difference equations (Q701338)

Let \(\{X_k\}\) be a process satisfying the stochastic difference equation \(X_k=A_kX_{k-1} +B_k\), \(k=1,2,\dots\), where \(\{A_k,B_k\}\) are i.i.d bivariate random variables. The authors define the sub-sampled series \(Y_k= X_{Mk}\), \(k=1,2, \dots,\) corresponding to a fixed systematic sampling interval \(M>1\). They discuss conditions for the existence of stationary solutions of the sub-sampled process \(\{Y_k\}\), and analyse the tail behavior of the stationary distribution of \(Y_k\) when the random vectors \(\{A_k,B_k\}\) have different tail behavior. Finally, the extremal behavior of the sub-sampled process \(\{Y_k\}\) is investigated. The results are used to the study of the tail and extremal behavior of sub-sampled bilinear processes and sub-sampled ARCH processes.