Asintoticità del numero di bocchi in partizioni casuali generate da un random seating plan

The aim of this thesis is to study the asymptotic behaviour of the random partition generated by a Random Seating Plan. Several papers have long dwelled on the analysis of various functionals of such partition and in the following chapters we will recall some important results on the number of blocks generated by the seating plan. In particular, opposite to the continuous time construction developed using renewal theory, an alternative discrete time construction will be presented, and some quantities of interest will be analyzed in detail. In the first chapter a brief review on the Dirichlet Process will be exposed: particular emphasis will be devoted to some key properties of the process, such as its conjugacy, its posterior mean, the predictive distribution and its stick-breaking construction. This last property in particular will be essential to introduce the study of random partitions presented in the following chapters. The second chapter is devoted to random partitions; in particular, the definition of Chinese Restaurant Process and Random Seating Plan will be given, together with the exchangeable partitions they produce and their probability functions. This well known results in fact are essential to shed light on the topic and several facts often represent key results for further studies. In the third chapter a summary of the work of Alexander Gnedin on the so called Bernoulli Sieve is presented. The main result presented here will be a Central Limit Theorem for the number of clusters generated by stick-breaking construction of the random partition using a sequence an independent sequence of random frequencies. Several tools of renewal theory will be used and sufficient conditions will be given for the convergence in distribution to a standard Gaussian limit. In the last chapter is exposed the discrete time representation of this construction. In particular, using the perspective on the increments of the Random Seating Plan studied by Nacu, a point process will be defined and some key features will be presented, such as the distribution of the time between the jumps for instance. Evidently, all of this work is a preliminary analysis on the topic and further interesting expansions could be made. Among those, it would be of major relevance a possible correspondence in discrete time with the Central Limit result presented in chapter 3.