  We consider a self-similar fragmentation process in which the generic
particle of size $x$ is replaced at probability rate $x^\alpha$, by its
offspring made of smaller particles, where $\alpha$ is some positive parameter.
The total of offspring sizes may be both larger or smaller than $x$ with
positive probability. We show that under certain conditions the typical size in
the ensemble is of the order $t^{-1/\alpha}$ and that the empirical
distribution of sizes converges to a random limit which we characterise in
terms of the reproduction law.
