Cramér's decomposition theorem

Cramér’s decomposition theorem for a normal distribution is a result of probability theory. It is well known that, given independent normally distributed random variables ξ₁, ξ₂, their sum is normally distributed as well. It turns out that the converse is also true. The latter result, initially announced by Paul Lévy, has been proved by Harald Cramér. This became a starting point for a new subfield in probability theory, decomposition theory for random variables as sums of independent variables (also known as arithmetic of probabilistic distributions).