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This paper is devoted to the study of the interaction between two distinctforms of non-stationary processes, which we will refer to as non-stationarityof first and second kind. The non-stationarity of first kind is caused bycriticality-generated events that we call crucial events. Crucial events signalergodicity breaking emerging from the interaction between the units of thecomplex system under study, indicating that the non stationarity of first kindhas internal origin. The non-stationarity of second kind is due to theinfluence on the system of interest of an environment changing in time, therebyimplying an external origin. In this paper we show that the non-stationarity offirst kind, measured by an inverse power law index {\mu} is characterized bysingularities at {\mu} = 2 and {\mu} = 3. We realize the interaction betweenthe non-stationarity of first kind and the non-stationarity of second kind witha model frequently adopted to study earthquakes, namely, a system ofmainshocks, assumed to be crucial events, generating a cascade of after-shockssimulating the changing in time environment. We prove that the after-shockssignificantly affects the detection of anomalous scaling, with this effectweakening as the value {\mu} approaches {\mu} = 2.5. We argue that this resultis a consequence of the fact that the states {\mu} = 2 and {\mu} = 3 are theborders between different statistical regimes, where a sort of phase transitionoccurs, with {\mu} = 2.5 being a state sufficiently far from both transitionregimes. We conclude this paper with the observation that the earthquakesshould be interpreted as resulting from the interaction between manygeophysical units generating criticality, with the non-stationary events ofsecond kind affecting conveniently short time regions between two consecutivecrucial events.

Influence of an environment changing in time on Crucial Events: the earthquake prototype