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For any positive integer $N$, the binomial$(N!,p)$ distribution has the following property: for any $1 \leq n \leq N$, there exist i.i.d. random variables $X_1,\ldots,X_n$ such that $X_1 + \cdots + X_n \sim {\rm binomial}(N!,p)$ (specifically, we take $X_1,\ldots,X_n$ to be i.i.d. binomial$(N!/n,p)$ rv's). It may be interesting to consider the following question: given $N \geq 3$, arbitrary but fixed, is there a continuous bounded distribution $\mu = \mu_N$ having the same property? (I stress: continuous and bounded.)

EDIT: Well, it turns out this is a really trivial problem, but worth remembering...; see my answer below.

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    Yep, just read it before you edited that message. So it's no good either. You got an interesting question there. +12010-12-05

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EDIT: NEW ANSWER

It is convenient to express the solution in terms of convolutions: for any distribution $\mu$, positive integer $N$, and $1 \leq n \leq N$, we have $ \mu ^{*N!} = \big(\mu ^{*(N!/n)} \big)^{*n}. $ So simple, yet instructive...

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    The fact concerning the singular distributions is interesting. Actually, thanks to your observation regarding the convolution of $N!$ uniform distributions, I noticed that the problem is trivial.2010-12-06
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OK, I think I've got one for you, but I have to admit it's a bit of a cheat. If you replace independence in your requirements by free independence then there exists a continuous bounded and infinitely divisible distribution, namely the Wigner semi-circle distribution.

That's the only example I can think of for now.

I also give this reference for details.

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    Well, understanding the basics requires some knowledge of functional analysis, especially Hilbert spaces, von Neumann algebras and C*-algebras. It has been developed by Dan Voiculescu as an alternative statistics theory, a bit like there are non-Euclidean geometries. I think there's also a third type of statistics, based on another kind of independence. But I forgot how it's called.2010-12-05