  For a subcritical Galton-Watson process $(\zeta_n)$, it is well known that
under an $X \log X$ condition, the quotient $P(\zeta_n > 0)/ E\zeta_n$ has a
finite positive limit. There is an analogous result for a (one-dimensional)
supercritical branching random walk: when $a$ is in the so-called subcritical
speed area, the probability of presence around $na$ in the $n$-th generation is
asymptotically proportional to the corresponding expectation. In Rouault (1993)
this result was stated under a natural $X \log X$ assumption on the offspring
point process and a (unnatural) condition on the offspring mean. Here we prove
that the result holds without this latter condition, in particular we allow an
infinite mean and a dimension $d \geq 1$ for the state-space. As a consequence
the result holds also for homogeneous fragmentations as defined in Bertoin
(2001), using the method of discrete-time skeletons; this completes the proof
of Theorem 4 in Bertoin-Rouault (2004 see math/PR/0409545). Finally, an
application to conditioning on the presence allows to meet again the
probability tilting and the so-called additive martingale.
