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Let $Z$ be a derived random variable given by some function $f$, i.e. $Z=f(X_1,\dots, X_n)$, where the $X_i\sim X$ are continuous/non-atomic and independently and identically distributed. What can we say about $X$?

The function $f$ is assumed to be non-constant and invariant under permutations of the coordinates. I have a specific class of examples in mind, where the $X_i$ are defined on a metric space and $f$ depends on the pairwise distances of the $X_i$.

Probably, finding out the distribution of $X$ is too difficult or even impossible in the general case. But what about some simpler properties, for example unimodality, symmmetry, the support, the moments or other summary statistics? Is there a general approach to this? Do you know any references?

Remark: In the special case of $Z=X_1+X_2$, the densities satisfy $\rho_Z=\rho_X*\rho_X$, so by the convolution theorem, there exists a formal solution $\rho_X=\mathcal F^{-1}(\sqrt {\mathcal F(\rho_Z)}$. Maybe there is a functional analytic framework which gives similar results for similar/different/arbitrary functions $f$?

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  • $\begingroup$ Do I understand correctly that you want to find the distribution of $X$ from the distribution of $Z$? $\endgroup$
    – joriki
    Commented Jan 16, 2020 at 11:28
  • $\begingroup$ @joriki Yes, but i doubt its possible to find the distribution in the general case. So rather, given the distribution of $Z$, I want to extract some information about $X$. $\endgroup$
    – klirk
    Commented Jan 16, 2020 at 14:47
  • $\begingroup$ You may want to add information in your statement, I feel like if $f$ is a constant, then you cannot say anything about $P_X$. Maybe something like $f(x_1,\dots, x_{i-1},\cdot,x_{i+1},\dots,x_n)$ is one to one. Also maybe something like $f$ is invariant to some permutation of the $x_i$s would make it easier. $\endgroup$
    – P. Quinton
    Commented Jan 20, 2020 at 22:16
  • $\begingroup$ @P.Quinton: I am looking for a general approach, so I did not want to put unnecessary restrictions on $f$. But your are right, we need some assumption. I will modify the question accordingly. I don't want to assume the 1-1 property though, as $X$ and $Z$ can be defined on different spaces. $\endgroup$
    – klirk
    Commented Jan 20, 2020 at 22:43
  • $\begingroup$ If $f(x_1,\dots,x_n)=1(x_1=x_2=\dots=x_n)$, and $X$ is a random variable over $[0,1]$, then suppose I tell you that $Z=1$ with probability $(1-\varepsilon)^n+\varepsilon^n$ for $\varepsilon\in [0,1]$. Then $X=1$ with probability $\varepsilon$ and $0$ with probability $1-\varepsilon$. Now if both $b$ and $a+b$ (with $b\neq 0$ )are in $[0,1]$ then $aX+b\in [0,1]$ and it leads to the same distribution of $Z$. $\endgroup$
    – P. Quinton
    Commented Jan 21, 2020 at 11:49

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