Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$


I have observations of $Z\_i$'s and thus can approximate the discrete pdf $f()$ which is the distribution of Z.


Thus: 


$f(x) = (g \ast g^\prime)(x) = \sum\_{d \in D} g(d) g((x-d)c)$


where $g^\prime(x) = g(xc)$


How to calculate $g()$ based on $f()$?



Deconvolution of sum of two random variables
I found this paper [Rates of convergence for constrained deconvolution problem](http://arxiv.org/pdf/math/0306237.pdf) which describes how to estimate the distribution of $Z$ from observations of a random process of the form $Z = \alpha X + \beta Y$.


The trick is to use the characteristic function of Z and it really works out quite nicely.


