Extremes of a class of deterministic sub-sampled processes with applications to stochastic difference equations

Let { X k } k ⩾ 1 be a stationary sequence of the form X k = ∑ j = 1 ∞ ∏ s = 1 j - 1 A k - s B k - j , where { A k , B k } are i.i.d. R + 2 -valued random pairs with some given joint distribution. For a strictly increasing subsequence { g ( k ) } , let Y k = X g ( k ) be the deterministic sub-sampled sequence. The aim of this paper is to look at the limiting form of certain empirical point processes induced by { Y k } for a specific class of deterministic sampling functions g ( · ) . Such asymptotic results will be useful in obtaining the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the limiting distribution of the maximum and its corresponding extremal index.