Statistical analysis of self-similar conservative fragmentation chains

We explore statistical inference in self-similar conservative fragmentation chains, when only (approximate) observations of the size of the fragments below a given threshold are available. This framework, introduced by Bertoin and Martinez, is motivated by mineral crushing in mining industry. The underlying estimated object is the step distribution of the random walk associated to a randomly tagged fragment that evolves along the genealogical tree representation of the fragmentation process. We compute upper and lower rates of estimation in a parametric framework, and show that in the non-parametric case, the difficulty of the estimation is comparable to ill-posed linear inverse problems of order 1 in signal denoising.