# Divisible Random Variables

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.

• The $m_i$s can be dependant on the $X_i$s? – Brendan McKay May 19 '12 at 11:36
• Actually, everything on the RHS is independent. Thanks. – Ben May 19 '12 at 14:01
• do you have any examples of such things ? – mike May 23 '12 at 23:13