Kernel based Dirichlet sequences

TL;DR

This paper introduces kernel-based Dirichlet sequences, extending classical Dirichlet processes by incorporating kernel functions, and establishes their properties, convergence behaviors, and conditions for discreteness or absolute continuity.

Contribution

It generalizes Dirichlet sequences using kernel functions, providing new theoretical properties, convergence results, and conditions for the nature of the limiting measure.

Findings

Sequences are exchangeable under certain kernel conditions.

Convergence in total variation of predictive distributions is established.

Conditions for the limit measure to be discrete, non-atomic, or absolutely continuous are provided.

Abstract

Let $X=(X_1,X_2,\ldots)$ be a sequence of random variables with values in a standard space $(S,\mathcal{B})$. Suppose \begin{gather*} X_1\sim\nu\quad\text{and}\quad P\bigl(X_{n+1}\in\cdot\mid X_1,\ldots,X_n\bigr)=\frac{\theta\nu(\cdot)+\sum_{i=1}^nK(X_i)(\cdot)}{n+\theta}\quad\quad\text{a.s.} \end{gather*} where $\theta>0$ is a constant, $\nu$ a probability measure on $\mathcal{B}$, and $K$ a random probability measure on $\mathcal{B}$. Then, $X$ is exchangeable whenever $K$ is a regular conditional distribution for $\nu$ given any sub-$\sigma$-field of $\mathcal{B}$. Under this assumption, $X$ enjoys all the main properties of classical Dirichlet sequences, including Sethuraman's representation, conjugacy property, and convergence in total variation of predictive distributions. If $\mu$ is the weak limit of the empirical measures, conditions for $\mu$ to be a.s. discrete, or a.s. non-atomic, or $\mu\ll\nu$ a.s., are provided. Two CLT's are proved as well. The first deals with stable convergence while the second concerns total variation distance.