  The detrended fluctuation analysis (DFA) is extensively useful in stochastic
processes to unveil the long-term correlation. Here, we apply the DFA to point
processes that mimick earthquake data. The point processes are synthesized by a
model similar to the Epidemic-Type Aftershock Sequence model, and we apply the
DFA to time series $N(t)$ of the point processes, where $N(t)$ is the
cumulative number of events up to time $t$. Crossover phenomena are found in
the DFA for these time series, and extensive numerical simulations suggest that
the crossover phenomena are signatures of non-stationarity in the time series.
We also find that the crossover time represents a characteristic time scale of
the non-stationary process embedded in the time series. Therefore, the DFA for
point processes is especially useful in extracting information of
non-stationary processes when time series are superpositions of stationary and
non-stationary signals. Furthermore, we apply the DFA to the cumulative number
$N(t)$ of real earthquakes in Japan, and we find a crossover phenomenon similar
to that found for the synthesized data.
