  Several recent works point out that the crowd of small unobservable
earthquakes (with magnitudes below the detection threshold $m_d$) may play a
significant and perhaps dominant role in triggering future seismicity. Using
the ETAS branching model of triggered seismicity, we apply the formalism of
generating probability functions to investigate how the statistical properties
of observable earthquakes differ from the statistics of all events. The ETAS
(epidemic-type aftershock sequence) model assumes that each earthquake can
trigger other earthquakes (``aftershocks''). An aftershock sequence results in
this model from the cascade of aftershocks of each past earthquake. The
triggering efficiency of earthquakes is assumed to vanish below a lower
magnitude limit $m_0$, in order to ensure the convergence of the theory and may
reflect the physics of state-and-velocity frictional rupture. We show that, to
a good approximation, the ETAS model is renormalized onto itself under what
amounts to a decimation procedure $m_0 \to m_d$, with just a renormalization of
the branching ratio from $n$ to an effective value $n(m_d)$. Our present
analysis thus confirms, for the full statistical properties, the results
obtained previously by one of us and Werner, based solely on the average
seismic rates (the first-order moment of the statistics). However, our analysis
also demonstrates that this renormalization is not exact, as there are small
corrections which can be systematically calculated, in terms of additional
contributions that can be mapped onto a different branching model (a new
relevant direction in the language of the renormalization group).
