Asymptotic regimes for the occupancy scheme of multiplicative cascades

In the classical occupancy scheme, one considers a fixed discrete probability measure ${\bf p}=(p_i: {i\in{\cal I}})$ and throws balls independently at random in boxes labeled by ${\cal I}$, such that $p_i$ is the probability that a given ball falls into the box $i$. In this work, we are interested in asymptotic regimes of this scheme in the situation induced by a refining sequence $({\bf p}(k) : k\in\N)$ of random probability measures which arise from some multiplicative cascade. Our motivation comes from the study of the asymptotic behavior of certain fragmentation chain

A phase transition for the heights of a fragmentation tree

We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of the partition process have cardinality less than a fixed integer. Our results may be applied to the study of certain random split trees

An Urn Model from Learning Theory

We present an urn model that is a variation of the classical occupancy model, and in which the balls are of two types (good and bad). We analyze the number of urns that contain more (or less) good balls than bad balls. We find Gaussian limiting distributions in the static case and convergence of the finite-dimensional distributions towards those of a Gaussian, non-Markov process in the dynamic case. c fl 1996 John Wiley & Sons, Inc. Keywords: Bessel functions, Exchangeable variables, Gaussian Process, Generating function, Urn models. 1. INTRODUCTION A. From learning theory to urn problems The original motivation of this investigation came from Computational Learning Theory [14]. During recent years, learning theory has paid a renewed attention to learning curves. Those curves monitor the improvement of the performance of the learner as she gets more information from her environment. Investigations based on * This research was supported by the ESPRIT Working Group RAND II and CNRS-GDR..