Ergodic theorems for stress release processes (Q809482)

The stress release processes are considered by a number of authors because of the seismological interest (importance). The processes present the regime of stress built up linearly and released stochastically with a risk function and jump distributions. The author mentions four different models (cases) of the processes which have been studied by various authors, \textit{D. Vere-Jones} [J. R. Stat. Soc., Ser. B 32, 1-45, 46-62 (1970; Zbl 0201.277)] was the first who introduced such models in the seismological context. The main purpose of this note is to obtain results on general stress release processes. In Section 2 the author studies the structure of the general stress release models, questions of existence, uniqueness and validity of the Kolmogorov equations. Section 3 contains the limit behaviour of Model 4. A structure of the processes is given and Harris' ergodicity and geometrical ergodicity of the processes are studied. In Section 4 he uses the method developed in Section 3 to study ergodic theorems for the other three cases.