Contributions to the Theory of Dirichlet Processes

Abstract

Consider a sample $X_1, \cdots, X_n$ from a Dirichlet process $P$ on an uncountable standard Borel space $(\mathscr{X}, \mathscr{A})$ where the parameter $\alpha$ of the process is assumed to be non-atomic and $\sigma$-additive. Let $D(n)$ be the number of distinct observations in the sample and denote these distinct observations by $Y_1, \cdots, Y_{D(n)}$. Our main results are (1) $D(n)/\log n \rightarrow_{\operatorname{a.s.}} \alpha(\mathscr{X}), n \rightarrow \infty$, and (2) given $D(n), Y_1, \cdots, Y_{D(n)}$ are independent and identically distributed according to $\alpha(\bullet)/\alpha(\mathscr{X})$. Result (1) shows that $\alpha(\mathscr{X})$ can be consistently estimated from the sample, and result (2) leads to a strong law for $\sum^{D(n)}_{i=1} Y_i/D(n)$.