Conditionally identically distributed species sampling sequences

TL;DR

This paper introduces Generalized Species Sampling Sequences, including the GOS class, and studies their convergence properties and partition lengths, extending dependence concepts beyond exchangeability with applications to urn schemes.

Contribution

It defines and analyzes Generalized Species Sampling Sequences and GOS, providing new convergence results and partition length laws under conditional identity in distribution.

Findings

Convergence of empirical and predictive means to Gaussian mixtures.

Stable and almost sure convergence results.

Strong law of large numbers and CLT for partition lengths.

Abstract

Conditional identity in distribution (Berti et al. (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper, a class of random sequences, called Generalized Species Sampling Sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, a class of generalized species sampling sequences that are conditionally identically distributed is introduced and studied: the Generalized Ottawa sequences (GOS). This class contains a '`randomly reinforced'' version of the P\'olya urn and of the Blackwell-MacQueen urn scheme. For the empirical means and the predictive means of a GOS, we prove two convergence results toward suitable mixtures of Gaussian distributions. The first one is in the sense of stable convergence and the second one in the sense of almost sure conditional convergence. In the last part of the paper we study the length of the partition induced by a GOS at time $n$, i.e. the random number of distinct values of a GOS until time $n$. Under suitable conditions, we prove a strong law of large numbers and a central limit theorem in the sense of stable convergence. All the given results in the paper are accompanied by some examples.