Timeline for Random partition of an interval – Dirichlet distributed?
Current License: CC BY-SA 4.0
13 events
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| Apr 4 at 19:29 | answer | added | Michael Hardy | timeline score: 0 | |
| Apr 4 at 18:39 | comment | added | Michael Hardy | Suppose $1\le k_1\le k_2\le \cdots \le k_\ell \le N$ are integers. Then the probability distribution of the vector of lengths of $$ [0, X_{(k_1)}], [X_{(k_1}, X_{(k_2)}], \ldots, [X_{(k_\ell)}, 1] $$ is $\operatorname{Dirichlet}(k_1,\,k_2-k_1,\,k_3-k_2,\,\ldots, \, N-k_\ell).$ I suspect I can prove this by paraphrasing Thomas Bayes's famous posthumous paper An Essay Toward Solving a Problem in the Doctrine of Chances, published in 1761. Bayes did the case $\ell=1,$ getting what today we call a Beta distribution. The Beta distribution is a special case of the Dirichlet distribution. | |
| Apr 4 at 12:40 | review | Close votes | |||
| Apr 9 at 3:03 | |||||
| Apr 4 at 12:09 | answer | added | van der Wolf | timeline score: 1 | |
| Mar 5 at 15:24 | comment | added | Dieter Kadelka | Did you take a look into \item {\bf Bertoin, J.:} The asymptotic behavior of fragmentation processes, J. European Math. Soc., 5 (2003) 395 - 416 and \item {\bf Bertoin, J.:} Different Aspects of a Model for Random Fragmentation Processes. ccsd-00005175, version 1, 7 Feb 2005, 26 S. | |
| Mar 5 at 13:59 | history | edited | gusl | CC BY-SA 4.0 |
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| Feb 29 at 2:15 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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| Feb 28 at 18:51 | comment | added | Jukka Kohonen | Pretty sure it has been published, many times. For example, Springer's Encyclopedia of Mathematics, Dirichlet distribution explicitly says the distribution is Dirichlet. (FWIW Wikipedia also says it, but without an inline reference.) | |
| Feb 28 at 2:30 | comment | added | Michael Hardy | I believe your guess is right, but it's hard to see this as research-level math even if it hasn't been published. Christian Remling's comment is terse but it looks like the right idea. I think he meant the whole vector of all $N$ order statistics rather than for just one value of $j.$ | |
| Feb 28 at 1:21 | review | Close votes | |||
| Mar 16 at 3:05 | |||||
| Feb 28 at 1:05 | comment | added | Christian Remling | Of course, that's assuming that the $X_j$ are independent; you obviously can't say anything in general. | |
| Feb 28 at 1:02 | comment | added | Christian Remling | This follows from the fact that the joint density of the order statistics $X_{(j)}$ is constant and the change of variable from the $x_{(j)}$ to the lengths has constant Jacobian (it's linear), so the probability is obtained by integrating a constant, which is exactly how the Dirichlet distribution with all parameters equal to $1$ works. | |
| Feb 27 at 23:21 | history | asked | gusl | CC BY-SA 4.0 |