On the maximum term of MA and Max-AR models with margins in Anderson's class

We consider several integer-valued stationary models of MA and max-AR type and study the limiting distribution of the maximum term after appropriate normalization. In particular, we consider marginal distributions which do not belong to the domain of attraction of any extreme value distribution but exhibit a quasi-stable limiting behavior in the sense of Anderson [J. Appl. Probab., 7 (1970), pp. 99–113] and therefore belong to the domain of attraction of a max-semistable law. Examples of such distributions are the negative binomial and the logarithmic distribution which are widely used to model real data applications. By verifying appropriate dependence conditions we obtain the limiting distribution of the maximum term for the models under consideration. Motivation comes from the analysis of the extremal behavior of integer-valued data requiring specific time series modeling.