The Tails of Probabilities Chosen From A Dirichlet Prior

Let $\alpha$ be a finite nonnull measure on $\mathbb{R}$, and let the random distribution function $F$ be distributed according to $\mathbb{D}_\alpha$, where $\mathbb{D}_\alpha$ is the Dirichlet process prior with parameter $\alpha$; see Ferguson (1973). This note points out that, almost surely, the tails of $F$ are much smaller than the tails of $\alpha$.