Stochastic processes of the urn type with convergent predictive distributions

In this work we propose a general class of stochastic processes with random reinforcement that are extensions of the celebrated Pòlya sequence by Blackwell and MacQueen [Ann. Stat. 1 (1973) 353--355]. The resulting randomly reinforced Pòlya sequences (RRPS) can be described as urn schemes with countable number of colors and general replacement rules. Under assumptions of conditional independence between reinforcement and observation, a RRPS becomes conditionally identically distributed (in the sense of [Ann. Probab. 32 (2004) 2029--2052]), and thus predictively convergent, in which case we show that it is asymptotically equivalent in law to an exchangeable species sampling sequence. This result has important implications on the generated random partition, which can be visualized as a weighted version of the Chinese Restaurant Process. We then provide complete distributional characterization of the predictive limit for the model with dichotomous reinforcements. Throughout the second part of the thesis, we consider an alternative specification of the replacement mechanism of a RRPS, whereby we deem some colors to be probabilistically dominant. In this situation the predictive and empirical distributions evaluated near the set of dominant colors both tend to 1. In fact, under some further restrictions on the reinforcement, the predictive and empirical distributions converge in the sense of almost sure weak convergence to one and the same random probability measure, whose mass is concentrated on the dominant set. As a consequence, the process becomes asymptotically exchangeable and its law -- directed by the above random measure, so that the data structure gets relatively sparse with time. The predictive limit for both models is generally unknown, however, so we derive central limit results, with which to approximate its distribution. The last chapter of the thesis is addressed towards applications of the RRPS, with the dominant-color model being considered in the context of clinical trials with response-adaptive design. Sections discussing uni- and multivariate extensions of the RRPS complete our study