Suppose I can write a positive, real valued random variable $$ X = m_1 X_1 + m_2 X_2,$$ where $m_1$ and $m_2$ are i.i.d, $X_1$ and $X_2$ are i.i.d and moreover, the $X_i$ are distributed like $X$. Also, assume that all the random variables have finite first and second moments ( $\mathbb{E} m_i =1/2$). Is this question studied at all? Do such random variables have to be 'nice'? In particular are there some weak conditions on $X$ under which $\frac{m_1 X_1}{X}$ is Dirichlet? To me this seems like a close analog of infinitely divisible random variables with very structured dependence but I haven't been able to find any references on it. Thanks for any help.
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$\begingroup$ The $m_i$s can be dependant on the $X_i$s? $\endgroup$– Brendan McKayCommented May 19, 2012 at 11:36
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$\begingroup$ Actually, everything on the RHS is independent. Thanks. $\endgroup$– BenCommented May 19, 2012 at 14:01
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$\begingroup$ do you have any examples of such things ? $\endgroup$– mikeCommented May 23, 2012 at 23:13
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