Suppose that the positive random variable $X$ is infinitely divisible and supported on $\mathbb R\_+$. Due to Lévy-Khintchine, its moment generating function then writes :


$$M(t) = \mathbb E\left(e^{tX}\right) = \exp\left\{\int\_{\mathbb R\_+} \left(e^{ty}-1\right)L(dy)\right\}.$$


**Question:** Is there a way, from a $n$-sample $X\_1,...X\_n$ of i.i.d. random variables, to estimate the (non-negative) Lévy measure $L$ ?



Estimation of Lévy measure of ID distribution
As is done in estimation of regression or pdf, you can parametrize the problem -- say by assuming that
$$\frac{L(dy)}{dy}=\sum\_{j=1}^k c\_j g\_j(y),$$
where the $g\_j$'s are known nonnegative functions and the $c\_j$'s are unknown nonnegative parameters. For instance, if you choose your $g\_j$'s to be the indicators of finite intervals or truncated normal densities (so that
$$g\_j(y)=\frac1{b\_j\sqrt{2\pi}}e^{-(y-a\_j)^2/(2b\_j^2)}$$
for some real $a\_j$'s and some positive real $b\_j$'s), then you can evaluate $\int\_{\mathbb R\_+} \left(e^{ty}-1\right)L(dy)$ explicitly. Then your moment generating function (mgf) will be explicitly determined by the $k$-tuple $c:=(c\_1,\dots,c\_k)$ of the parameters: $M=M\_c$.


Then you can fit the parametric mgf $M\_c(t)$ to the
empirical mgf
$$\hat M(t):=\frac1n\,\sum\_{i=1}^n e^{tX\_i}$$
to the true mgf $M(t)$ --- say, by minimizing
$$\int\_{\mathbb R}(M\_c(t)-\hat M(t))^2 w(t)\,dt$$
in $c=(c\_1,\dots,c\_k)$ for some nonnegative weight function $w$.


