Recent advances on the occupancy problem

From a classical point of view, the occupancy problem (also known as the urn scheme) is to describe the spread (or repartition) of a random sample over its support when it is drawn from a discrete distribution. In this talk we will start by briefly describing the problem and state a few well known results concerning the asymptotic behaviour and concentration of the so called occupancy probabilities. In the second part of the talk, we will present new results concerning finite sample (upper and lower) bounds for the occupancy probabilities. Finally, the third and last part of the talk will discuss some results and perspectives outside of the i.i.d. and discrete context: arbitrary distributions in a metric space and Markov chains.