TEST https://doi.org/10.1007/s11749-017-0573-z ORIGINAL PAPER The max-INAR(1) model for count processes Manuel G. Scotto 1 · Christian H. Weiß 2 · Tobias A. Möller 2 · Sónia Gouveia 3 Received: 4 July 2017 / Accepted: 28 November 2017 © Sociedad de Estadística e Investigación Operativa 2017 Abstract This paper proposes a discrete counterpart of the conventional max- autoregressive process of order one. It is based on the so-called binomial thinning operator and driven by a sequence of independent and identically distributed non- negative integer-valued random variables with either regularly varying right tail or exponential-type right tail. Basic probabilistic and statistical properties of the process are discussed in detail, including the analysis of conditional moments, transition prob- abilities, the existence and uniqueness of a stationary distribution, and the relationship between the observations’ and innovations’ distribution. We also provide conditions on the marginal distribution of the process to ensure that the innovations’ distribution exists and is well defined. Several examples of families of distributions satisfying such conditions are presented, but also some counterexamples are analyzed. Furthermore, This research was supported by the German Academic Exchange Service (DAAD) and the Fundação para a Ciência e a Tecnologia (FCT), under the program “Ações Integradas Luso-Alemãs” and the Grants 57212119 and A-38/16. Manuel Scotto also acknowledges the Project UID/Multi/04621/2013. S. Gouveia acknowledges the postdoctoral Grant by FCT (ref. SFRH/BPD/87037/2012). This work was also partially supported by the Portuguese FCT, with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020—within IEETA/UA Project UID/CEC/00127/2013 (Instituto de Engenharia Electrónica e Informática de Aveiro, IEETA/UA, Aveiro) and CIDMA/UA Project UID/MAT/04106/2013 (Centro de Investigação e Desenvolvimento em Matemática e Aplicações, CIDMA/UA, Aveiro). B Manuel G. Scotto manuel.scotto@tecnico.ulisboa.pt 1 Departamento de Matemática and CEMAT, IST, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal 2 Department of Mathematics and Statistics, Helmut Schmidt University, Hamburg, Germany 3 Institute of Electronics and Informatics Engineering of Aveiro (IEETA) and Center for R&D in Mathematics and Applications (CIDMA), Universidade de Aveiro, Aveiro, Portugal 123