In many fields of applied physics, the phenomenology of the space-time phenomena to be understood (in general for prediction purposes) may be described in the following most simple way: events with random common positive amplitude occur randomly in time according to a continuous time random walk (CTRW) model; the prerequisite is therefore a statistical model for both the amplitude and inter-arrival times between events, here assumed mutually independent. Special attention is paid here to CTRW for which both amplitude and holding time have infinite mean value (the extreme and rare hypothesis). Such processes and their limiting version arise in particular as inverses of processes with stationary independent increments of special interest (chiefly related to the Lévy stable subordinator).

Among other related models, we investigate here some properties of this CTRW in situations where the occurrence of events is modelled by a discrete inverse-Linnik process which shares the rare event hypothesis; this class derives (statistically) its importance from its close relationship to many other meaningful processes such as the Lévy, gamma and Mittag-Leffler ones. Physically, Linnik and inverse-Linnik processes appear as a recurrent paradigm in relaxation theory of condensed matter. The limit laws for cumulative Linnik sequences and their time to failure are finally discussed.