Cramér's theorem

The set of Cramér ( after the Swedish mathematician Harald Cramér ) is the inverse of the well-known statement that the sum of independent normally distributed random variables is again normally distributed.

Set of Cramér

Is a normally distributed random variable, the sum of two independent random variables and, then the summands and also normally distributed.

A normal random variable can thus decompose only in normally distributed independent summands.

Note also the " counter- statement" of the central limit theorem, according to which the sum of a large number of independent not necessarily normally distributed summands is approximately normally distributed.

The set of Cramér has a certain stability towards small deviations: If the sum is approximately normally distributed ( in a sense ), then there are also the summands.

The theorem was originally formulated by Paul Lévy, but shortly afterwards proved by Harald Cramér. It is therefore sometimes referred to as a set of Lévy- Cramér, but this can lead to confusion with other sentences that name.

Sketch of proof

The proof can be very elegant result by application of analytical properties of characteristic functions: From the decomposition follows for the corresponding characteristic functions. The function is an entire function of growth order 2 without zeros, so the factors are also entire functions with growth order at most 2 It follows (using the example of the first factor ) the representation. From elementary properties of characteristic functions it finally follows the presentation, so that the characteristic function of a normal random variable with parameters and.

This proof sketch demonstrates very well the interaction of different mathematical disciplines, in this case the stochastic and classical function theory.