Suppose that I have iid random variables $\mathrm U_n \sim \mathrm U(0,1)$. Then, for $\mathrm Y_m$ defined as,

$$\mathrm Y_m = \min_{n \in [1,m]} \mathrm U_n$$

it is easy to compute $\mathbb{E}[\mathrm Y_m] = \frac1{m+1}$.

Now consider a compact convex region $\mathcal{C}$ in higher dimensions $\mathbb{R}^k$ and let $\mathbf p_n$ be fixed points in $\mathcal C$ where $\mathbf p_n$ may lie on the boundary of $\mathcal C$. Let $\mathrm D_n$ denote iid random variables in $\mathcal C$ such that $\mathbb P_{\mathrm D_n}(S) \geq \kappa \frac{\mathrm{vol}(S)}{\mathrm{vol}(\mathcal C)}~\forall S\subseteq \mathcal C$ for some
fixed $\kappa < 1$. Thus $\mathrm D_n$ denotes a "skewed" uniform distribution of points in $\mathcal C$.

Suppose that $\mathrm U_n$ now corresponds to the distance of $\mathrm D_n$ from $\mathbf p_n$ ie. $\mathrm U_n \triangleq ||\mathrm D_n - \mathbf p_n||$. Can someone point me in the right direction to find upper bounds on $\mathbb E[\mathrm Y_n]$ in such cases ( or direct me to references ) ?


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