Is there a monotone coupling of Dirichlet random variables? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:03:58Z http://mathoverflow.net/feeds/question/81551 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81551/is-there-a-monotone-coupling-of-dirichlet-random-variables Is there a monotone coupling of Dirichlet random variables? sbacallado 2011-11-21T20:50:04Z 2011-11-24T19:22:58Z <p>Let <code>$X=(X_1,X_2,X_3)\sim \text{Dirichlet}(a_1,a_2,a_3)$</code> and <code>$Y=(Y_1,Y_2,Y_3)\sim \text{Dirichlet}(a_1+b_1,a_2+b_2,a_3)$</code>, where all <code>$a_i$</code> and <code>$b_i$</code> are positive. Is there a natural coupling between $X$ and $Y$ such that <code>$X_1\geq Y_1$</code> and <code>$X_2\geq Y_2$</code> with probability 1? </p> <p>The following coupling does not guarantee this property: Define independent random variables <code>$G_{c}\sim \text{Gamma}(c)$</code> for <code>$c\in\{a_1,a_2,a_3,b_1,b_2\}$</code>, and let </p> <p><code>$$X_i = \frac{G_{a_i}}{G_{a_1}+G_{a_2}+G_{a_3}}$$</code> </p> <p>and</p> <p><code>$$Y_i = \frac{G_{a_i}+G_{b_i}\mathbb{I}(i\in\{1,2\})}{G_{a_1}+G_{b_1}+G_{a_2}+G_{b_2}+G_{a_3}}.$$</code></p> http://mathoverflow.net/questions/81551/is-there-a-monotone-coupling-of-dirichlet-random-variables/81829#81829 Answer by Omer for Is there a monotone coupling of Dirichlet random variables? Omer 2011-11-24T19:12:45Z 2011-11-24T19:22:58Z <p>This is not always possible.</p> <p>Fix $a_1,a_2,a_3,b_1$. As $b_2\to\infty$, we have $Y_1\to0$ in probability, so it is not stochastically larger than $X_1$. </p> <p>A necessary condition is domination of the expectations, namely $\frac{a_i+b_i}{\sum a_j+b_j} \ge \frac{a_i}{\sum a_j}$ for $i=1,2$, but this is not sufficient either. If $b_1/a_1=b_2/a_2$ and $b_i\to\infty$ then the $Y_i$'s converge to constants in probability.</p>