Sampling from Finite Random Partitions

Abstract

In this manuscript, finite random partition models of the interval are first considered. Next, sampling problems arising in this context are addressed: throw at random k points on any such randomly broken interval. Does the k-sample contain two or more fragments of the same type? Have all fragments been visited or are there any undiscovered ones left? We investigate the random counterpart of these questions which, for deterministic partitions, are known as Feller''s birthday and coupon collector sampling problems. We show that computations are quite explicit when considering a Dirichlet random breaking-stick scheme. Also, the problem of counting the number of fragments in the k-sample with i representatives (the fragments'' vector count) is addressed, leading to a Ewens sampling formula for finite random partitions. To this end, some connections of the Ewens'' problem with the birthday and coupon collector''s ones are exploited. At last, simple illustrative examples are supplied which highlight the main differences, from the sampling point of view, between the symmetric deterministic and random uniform partitions.