Cramér's decomposition theorem

Cramér’s decomposition theorem for a normal distribution is a result in the mathematical theory of probability. 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,^[1] has been proved by Harald Cramér.^[2] 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).^[3]

The precise statement of the theorem

Let a random variable ξ be normally distributed and admit a decomposition as a sum ξ = ξ₁ + ξ₂ of two independent random variables. Then the summands ξ₁ and ξ₂ are normally distributed as well.

One proof of Cramér's decomposition theorem uses the theory of entire functions.

See also

• Raikov's theorem: Similar result for Poisson distribution.