  Motivated by its potential application to earthquake statistics, we study the
exactly self-similar branching process introduced recently by Vere-Jones, which
extends the ETAS class of conditional branching point-processes of triggered
seismicity. One of the main ingredient of Vere-Jones' model is that the power
law distribution of magnitudes m' of daughters of first-generation of a mother
of magnitude m has two branches m'<m with exponent beta-d and m'>m with
exponent beta+d, where beta and d are two positive parameters. We predict that
the distribution of magnitudes of events triggered by a mother of magnitude $m$
over all generations has also two branches m'<m with exponent beta-h and m'>m
with exponent beta+h, with h= d \sqrt{1-s}, where s is the fraction of
triggered events. This corresponds to a renormalization of the exponent d into
h by the hierarchy of successive generations of triggered events. The empirical
absence of such two-branched distributions implies, if this model is seriously
considered, that the earth is close to criticality (s close to 1) so that beta
- h \approx \beta + h \approx \beta. We also find that, for a significant part
of the parameter space, the distribution of magnitudes over a full catalog
summed over an average steady flow of spontaneous sources (immigrants)
reproduces the distribution of the spontaneous sources and is blind to the
exponents beta, d of the distribution of triggered events.
