It comes to me that this kind of thing can be used in gene introgression models, as I heard a very intersting lecture yesterday. I got an iteration equation for this. But I believe there should be an analytical general results.
I have thought about this problem too, in the context of multi-target tracking - what is the distribution of the distance from any point in the measurement space to the nearest false detection, if each false detection is uniformly distributed over the measurement space? I think as the volume of the measurement space and the number of false detections increase, the volume enclosed by that distance is distributed as a gamma distribution, with the decay parameter determined by the false detection density. The result can be derived using (nearest/first) order statistics. I can't remember the details, but you could have a look in the attached paper if you are interested.
https://www.researchgate.net/publication/262056247_Powerful_Test_Statistic_for_Track_Management_in_Clutter?ev=prf_pub
Article Powerful Test Statistic for Track Management in Clutter
If I understand your question correctly this could be an old problem, maybe this helps: https://en.wikipedia.org/wiki/Dirichlet_process#The_stick-breaking_process
There are quite a few links at the end of the page to papers and sites about the Dirichlet process.
Thanks for the above two comments. I did know it gets involved with Dirichlet distribution, but wanted to know whether there is some analytical results for this question. One of my friends sent me an answer, I'll have a read and check with it, may post it here. Thank you again.
I've got a quite easy way to give the answer: for an arbitary value x, the probability that the largest piece exceeds it equals
the probility that 1 piece exceeds it,minus the probability that 2 pieces exceeds it, plus the probability that 3 pieces exceeds it,...
From mathematical point of view, it is only a De Morgan Equation, how stupid I was.... Thanks everybody.
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