Divisibility and the continuous time limit

Divisibility

We have discussed in the previous chapter how the sum of two iid random variables can be computed if the distribution of these two variables is known. One can ask the opposite question: knowing the probability distribution of a variable X, is it possible to find two iid random variables such that X = δX1 + δX2, or more precisely such that the distribution of X can be written as the convolution of two identical distributions. If this is possible, the variable X is said to be divisible.

We already know cases where this is possible. For example, if X has a Gaussian distribution of variance σ2, one can choose X1 and X2 to be have Gaussian distribution of variance σ2/2. More generally, if X is a Lévy stable random variable of parameter aµ (see Eq. (1.20)), then X1 and X2 are also Lévy stable random variables with parameter aµ/2. However, all random variables are not divisible: as we show below, if X has a uniform distribution in the interval [a, b], it cannot be written as X = δX1 + δX2 with δX1, δX2 iid.