Davis, Richard A.; Resnick, Sidney I. Basic properties and prediction of max-ARMA processes. (English) Zbl 0716.62098 Adv. Appl. Probab. 21, No. 4, 781-803 (1989). A max-autoregressive moving average (MARMA(p,q)) process \(X_ t\) satisfies the recursion \[ X_ t=\phi_ 1X_{t-1}\vee...\vee \phi_ pX_{t-p}\vee Z_ t\vee \theta_ 1Z_{t-1}\vee...\vee \theta_ qZ_{t-q}, \] where \(\phi_ i,\theta_ j\geq 0\), and \(Z_ t\) is i.i.d. with distribution function \(\exp (-\sigma x^{-1})\) for \(x\geq 0\), \(\sigma >0\). Such processes have finite-dimensional distributions which are max-stable. Necessary and sufficient conditions for existence of a stationary solution to the MARMA recursion are provided. The optimality criterion is designed to minimize the probability of large errors. Most of results remain valid for the case when the distribution of \(Z_ 1\) is only in the domain of attraction of \(\exp (-\sigma x^{-1})\). Reviewer: M.P.Mokljacuk Cited in 3 ReviewsCited in 54 Documents MSC: 62M20 Inference from stochastic processes and prediction 60G35 Signal detection and filtering (aspects of stochastic processes) 93E10 Estimation and detection in stochastic control theory Keywords:max-autoregressive moving average process; max-stable processes; prediction; MARMA processes; max-linear; ARMA processes; causality; domain of attraction of extreme value distribution; Necessary and sufficient conditions; existence of a stationary solution; MARMA recursion; probability of large errors PDFBibTeX XMLCite \textit{R. A. Davis} and \textit{S. I. Resnick}, Adv. Appl. Probab. 21, No. 4, 781--803 (1989; Zbl 0716.62098) Full Text: DOI Link