## Dirichlet process, uniform “confidence bands”

In Stigler's (1977) Fractional order statistics, with applications,'' he says (page 545) that he considers a special case of Ferguson's (1973) Dirichlet process (DP). Specifically, 1) the measurable space is just the unit interval [0,1] with Borel sets, and 2) instead of random probability measure $P(\cdot)$, we only need the random distribution function $F(t)=P([0,t])$. Stigler says the DP is indexed by measure $\nu(\cdot)$, which for my case is Lebesgue measure times a constant $n$ (i.e., $n$ times the length of the interval), and writes $F\sim\textrm{Dir}(\nu)$.

Given this random distribution function $F(\cdot)$ with support $[0,1]$, is there an analytic expression for functions $\ell(x)$ and $u(x)$ such that there is $1-\alpha$ probability (e.g., 95%) that $\ell(x)\le F(x)\le u(x)$ for all $x\in[0,1]$?

Note: I realize $\ell(x)$ and $u(x)$ are not uniquely defined (e.g., could find some $u(x)$ given $\ell(x)=0$, or alternatively some $\ell(x)$ given $u(x)=1$, etc.), so if there is a convention for how to restrict the problem to a unique solution, that would also be of interest.

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