Is the infimum of the Ky Fan metric achieved? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:33:03Z http://mathoverflow.net/feeds/question/35695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35695/is-the-infimum-of-the-ky-fan-metric-achieved Is the infimum of the Ky Fan metric achieved? Byron Schmuland 2010-08-15T22:53:00Z 2011-02-24T18:53:47Z <p>Consider the probability space $(\Omega, {\cal B}, \lambda)$ where $\Omega=(0,1)$, ${\cal B}$ is the Borel sets, and $\lambda$ is Lebesgue measure.</p> <p>For random variables $W,Z$ on this space, we define the Ky Fan metric by</p> <p>$$\alpha(W,Z) = \inf \lbrace \epsilon > 0: \lambda(|W-Z| \geq \epsilon) \leq \epsilon\rbrace.$$</p> <p>Convergence in this metric coincides with convergence in probability.</p> <p>Fix the random variable $X(\omega)=\omega$, so the law of $X$ is Lebesgue measure, that is, ${\cal L}(X)=\lambda$.</p> <blockquote> <p><b> Question:</b> For any probability measure $\mu$ on $\mathbb R$, does there exist a random variable $Y$ on $(\Omega, {\cal B}, \lambda)$ with law $\mu$ so that $\alpha(X,Y) = \inf \lbrace \alpha(X,Z) : {\cal L}(Z) = \mu\rbrace$ ?</p> </blockquote> <p><em>Notes:</em></p> <ol> <li><p>By Lemma 3.2 of <a href="http://arxiv.org/PS_cache/math/pdf/0601/0601524v1.pdf" rel="nofollow">Cortissoz</a>, the infimum above is $d_P(\lambda,\mu)$: the L&#233;vy-Prohorov distance between the two laws.</p></li> <li><p>The infimum is achieved if we allowed to choose both random variables. That is, there exist $X_1$ and $Y_1$ on $(\Omega, {\cal B}, \lambda)$ with ${\cal L}(X_1) = \lambda$, ${\cal L}(Y_1) = \mu$, and $\alpha(X_1,Y_1) = d_P(\lambda,\mu)$. But in my problem, I want to fix the random variable $X$.</p></li> <li><p><em> Why the result may be true: </em> the space $L^0(\Omega, {\cal B}, \lambda)$ is huge. There are lots of random variables with law $\mu$. I can't think of any obstruction to finding such a random variable.</p></li> <li><p><em> Why the result may be false: </em> the space $L^0(\Omega, {\cal B}, \lambda)$ is huge. A compactness argument seems hopeless to me. I can't think of any construction for finding such a random variable. </p></li> </ol> http://mathoverflow.net/questions/35695/is-the-infimum-of-the-ky-fan-metric-achieved/38477#38477 Answer by has2 for Is the infimum of the Ky Fan metric achieved? has2 2010-09-12T14:47:22Z 2010-09-12T14:47:22Z <p>Because what follows doesn't fit in a comment, I write it here as an answer; but they are merely comments. After computing the minimizer for several simple distributions, my impression is that the answer to this question is yes, and there will be many minimizers.</p> <p>Intuitively, it seems possible to build an optimizer as follows: we are given the law $\mu$ and we would like to find a function $f$ such that 1) the distribution of $f$ is $\mu$ and 2) $\alpha(f,X)$ is minimum. Let $\epsilon > 0$ be this minimum. Let $F(x) = \mu( (-\infty, x])$, i.e., $F$ is the distribution function associated with $\mu$. Let $G$ be the inverse function of $F$: $G(x) \doteq \inf\{y: F(y) \ge x \}$. By its definition $G$'s distribution is $\mu$. Draw the graphs of the functions $l(x) = x + \epsilon$ and $u(x) = x -\epsilon$ around the graph of the function $X(x) = x$. To get the minimizer, one cuts the graph of $G$ into $n$ small pieces with lines parallel to the $x$ axis and shifts around the pieces along these lines so that they lie between the graphs of $l$ and $u$ as much as possible. As the number of pieces increase and their size decreases you would expect this to converge to a function that is the desired minimizer. The result will depend on the particulars of this process.</p> <p>As to non-uniqueness: suppose $f$ is a minimizer. Denote with $E$ the subset of $[0,1]$ over which $f$ differs from $X$ by at least $\epsilon$. The values that $f$ takes over $E$ can be freely permuted without affecting the distribution and the distance between $f$ and $X$. So there will be infinitely many minimizers, when there is one.</p> http://mathoverflow.net/questions/35695/is-the-infimum-of-the-ky-fan-metric-achieved/43716#43716 Answer by Jon Bannon for Is the infimum of the Ky Fan metric achieved? Jon Bannon 2010-10-26T19:58:08Z 2010-10-26T19:58:08Z <p>This probably helps not at all, but I saw you were interested in Ky-Fan metric, and friends of mine have looked at these in a noncommutative setting in which there are some "extreme value" properties. Maybe there's something useful in there for you: <a href="http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.4239v3.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.4239v3.pdf</a></p>