Infinite divisibility of random variables

Question

A random variable $X$ is infinitely divisible if for any positive integer $k$, there exist $k$ i.i.d. random variables $Y_1,...,Y_k$ such that $\sum_{i=1}^k Y_i = X$. Given the PDF of an arbitrary unbounded random variable, how can we determine if it is infinitely divisible? Assuming the random variable $X$ is in fact infinitely divisible, is there an algorithm to determine the distribution of $Y_i$?

Answer 1

It can be extremely difficult to say whether a given distribution is infinitely divisible or not. Infinitely divisible distribution are studied using characteristic functions. If you know that $X$ is Infinitely divisible then the characteristic function of $Y_i$ is the n-th root of the characteristic function of $X$ but you have to define the n-th root carefully. The distribution of $Y_i$ can be written down explicitly only in very few cases.