TITLE: Deconvolution of sum of two random variables
QUESTION [7 upvotes]: 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 $f()$?

REPLY [3 votes]: I found this paper Rates of convergence for constrained deconvolution problem 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.