doi:10.1007/s11009-018-9663-yengBadía, F.G.Mercier, S.Sangüesa, C.Extensions of the Generalized Pólya ProcessART-2019-107580A new self-exciting counting process is here considered, which extends the generalized Pólya process introduced by Cha (Adv Appl Probab 46:1148–1171, 2014). Contrary to Cha’s original model, where the intensity of the process (linearly) increases at each jump time, the extended version allows for more flexibility in the dependence between the point-wise intensity of the process at some time t and the number of already observed jumps. This allows the “extended Pólya process” to be appropriate, e.g., for describing successive failures of a system subject to imperfect but effective repairs, where the repair can lower the intensity of the underlying counting process. Probabilistic properties of the new process are studied (construction from a homogeneous pure-birth process, conditions of non explosion, computation of distributions, convergence of a sequence of such processes, ..) and its connection with Generalized Order Statistics is highlighted. Positive dependence properties are next explored. Finally, the maximum likelihood method is considered in a parametric setting and tested on a few simulated data sets, to highlight the practical use of the new process in an application context.2019http://zaguan.unizar.es/record/14816610.1007/s11009-018-9663-yhttp://zaguan.unizar.es/record/148166oai:zaguan.unizar.es:148166info:eu-repo/grantAgreement/ES/DGA/E64info:eu-repo/grantAgreement/ES/MINECO-FEDER/MTM2015-63978-PMETHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 21 (2019), 1057 – 1085All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/openAccess