Kernel based Dirichlet sequences

Abstract

Let $X=(X_1,X_2,\dots )$ be a sequence of random variables with values in a standard space $(S,\mathcal{B})$. Suppose

$$X_1\sim \mathrm{\nu }\text{and}P(X_{n+1}\in \cdot \mid X_1,\dots ,X_n)=\frac{\mathit{\theta }\mathrm{\nu }(\cdot )+{\sum }_{i=1}^{n}K(X_i)(\cdot )}{n+\mathit{\theta }}\text{a.s.}$$

where $\mathit{\theta }>0$ is a constant, ν 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 ν given any sub-σ-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 μ is the weak limit of the empirical measures, conditions for μ to be a.s. discrete, or a.s. non-atomic, or $\mathrm{\mu }\ll \mathrm{\nu }$ a.s., are provided. Two CLT’s are proved as well. The first deals with stable convergence while the second concerns total variation distance.