Median for metric spaces
Volume 28 / 2001
Applicationes Mathematicae 28 (2001), 191-209
MSC: 60B05, 60G48, 62F10.
DOI: 10.4064/am28-2-6
Abstract
We consider a Köthe space $({\Bbb E}, \| \cdot \| _{{\Bbb E}})$ of random variables (r.v.) defined on the Lebesgue space $([0,1], {\bf B},\lambda )$. We show that for any sub-$\sigma $-algebra $\mathscr F$ of ${\bf B}$ and for all r.v.'s $X$ with values in a separable finitely compact metric space $(M,d)$ such that $d(X, x)\in {\Bbb E}$ for all $x\in M$ (we then write $X\in {\Bbb E}(M)$), there exists a median of $X$ given $\mathscr F$, i.e., an $\mathscr F$-measurable r.v. $Y\in {\Bbb E}(M)$ such that $\| d(X,Y)\| _{{\Bbb E}} \leq \| d(X,Z)\| _{{\Bbb E}}$ for all $\mathscr F$-measurable $Z$. We develop the basic theory of these medians, we show the convergence of empirical medians and we give some applications.
