On Linnik's continuous-time random walks

In many fields of applied physics, the phenomenology of the space-time phen omena to be understood tin general for prediction purposes) may be describe d in the following most simple way: events with random common positive ampl itude occur randomly in time according to a continuous time random walk (CT RW) model; the prerequisite is therefore a statistical model for both the a mplitude and inter-arrival times between events, here assumed mutually inde pendent. Special attention is paid here to CTRW for which both amplitude an d holding lime have infinite mean value (the extreme and rare hypothesis). Such processes and their limiting version arise in particular as inverses o f processes with stationary independent increments of special interest (chi efly related to the Levy stable subordinator). Among other related models, we investigate here some properties of this CTR W in situations where the occurrence of events is modelled by a discrete in verse-linnik process which shares the rare event hypothesis; this class der ives (statistically) its importance from its close relationship to many oth er meaningful processes such as the Levy, gamma and Mittag-Leffler ones. Ph ysically, Linnik and inverse-linnik processes appear as a recurrent paradig m in relaxation theory of condensed matter. The limit laws for cumulative L innik sequences and their time to failure are finally discussed.