An Estimate of the Compounding Distribution of a Compound Poisson Distribution

The distribution of a random variable $X$ is called a compound Poisson distribution if
$${\mathbf P}\{X=n\}= \int_0^\infty{\frac{{\lambda^n}}{{n1}}}\varepsilon^{-\lambda}dG(\lambda),$$
where $n=0,1,2,\dots$ and $G(\lambda)$ is a distribution function (weight function) such that $G(+0)=0$. Let $X_1,\dots,X_N$ be mutually independent random variables which obey a compound Poisson distribution. The paper establishes a connection between the moment problem and the problem of evaluating the weight function $G(\lambda )$; an algorithm is constructed which allows one to construct a sampling estimate $\hat G_N(\lambda)$ which depends only on $X_1, \cdots,X_N$ and $\lambda$; if $N\to\infty$, then $\hat G_N(\lambda)$ converges weakly to the unknown weight function $G(\lambda)$ with probability $1$.