In this paper we establish an extension of the method of approximating optimal discrete-time stopping problems by related limiting stopping problems for Poisson-type processes. This extension allows us to apply this method to a larger class of examples, such as those arising, for example, from point process convergence results in extreme value theory. Furthermore, we develop new classes of solutions of the differential equations which characterize optimal threshold functions. As a particular application, we give a fairly complete discussion of the approximative optimal stopping behavior of independent and identically distributed sequences with discount and observation costs.

A max-autoregressive moving average (MARMA(p, q)) process {Xt} satisfies the recursion for all t where φ i, , and {Zt} is i.i.d. with common distribution function Φ1,σ (X): = exp {–σ x–1} for . Such processes have finite-dimensional distributions which are max-stable and hence are examples of max-stable processes. We provide necessary and sufficient conditions for existence of a stationary solution to the MARMA recursion and we examine the reducibility of the process to a MARMA(p′, q′) with p′ <p or q′ < q. After introducing a natural metric between two jointly max-stable random variables, we consider the prediction problem for MARMA processes. Assuming that X1, …, Xn have been observed, we restrict our class of predictors to be max-linear, i.e. of the form , and find b1, …, bn to minimize the distance between this predictor and Xn+k for k 1. The optimality criterion is designed to minimize the probability of large errors and is similar in spirit to the dispersion criterion adopted in Cline and Brockwell (Stoch. Proc. Appl. 19 (1985), 281-296) for the prediction of ARMA processes with stable noise. Most of our results remain valid for the case when the distribution of Z1 is only in the domain of attraction of Φ1,σ. In addition, we give a naive estimation procedure for the φ 's and the θ 's which, with probability 1, identifies the true parameter values exactly for n sufficiently large.