  This paper is devoted to the study of the interaction between two distinct
forms of non-stationary processes, which we will refer to as non-stationarity
of first and second kind. The non-stationarity of first kind is caused by
criticality-generated events that we call crucial events. Crucial events signal
ergodicity breaking emerging from the interaction between the units of the
complex system under study, indicating that the non stationarity of first kind
has internal origin. The non-stationarity of second kind is due to the
influence on the system of interest of an environment changing in time, thereby
implying an external origin. In this paper we show that the non-stationarity of
first kind, measured by an inverse power law index {\mu} is characterized by
singularities at {\mu} = 2 and {\mu} = 3. We realize the interaction between
the non-stationarity of first kind and the non-stationarity of second kind with
a model frequently adopted to study earthquakes, namely, a system of
mainshocks, assumed to be crucial events, generating a cascade of after-shocks
simulating the changing in time environment. We prove that the after-shocks
significantly affects the detection of anomalous scaling, with this effect
weakening as the value {\mu} approaches {\mu} = 2.5. We argue that this result
is a consequence of the fact that the states {\mu} = 2 and {\mu} = 3 are the
borders between different statistical regimes, where a sort of phase transition
occurs, with {\mu} = 2.5 being a state sufficiently far from both transition
regimes. We conclude this paper with the observation that the earthquakes
should be interpreted as resulting from the interaction between many
geophysical units generating criticality, with the non-stationary events of
second kind affecting conveniently short time regions between two consecutive
crucial events.
