Cramer's theorem (normal distribution)
The Cramér's theorem (after the Swedish mathematician Harald Cramér ) is the inverse of the famous statement that the sum of independent normally distributed random variables is normally distributed again.
Cramer's theorem
If a normally distributed random variable is the sum of two independent random variables and , then the summands and are also normally distributed. A normally distributed random variable can only be broken down into normally distributed, independent summands.
Please also note 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.
Cramér's theorem has a certain stability towards small deviations : If the sum (in a certain sense) is approximately normally distributed, then so are the summands.
The sentence was originally formulated by Paul Lévy , but was only proven shortly afterwards by Harald Cramér. It is therefore sometimes referred to as the Lévy-Cramér sentence, but this can lead to confusion with other sentences of this name.
Evidence sketch
The proof can be elegantly carried out by using analytical properties of characteristic functions : From the decomposition it follows for the associated characteristic functions . The function is a whole function of the growth order 2 without zeros, therefore the factors are also whole functions with a growth order at most 2. The representation follows (using the example of the first factor) . The representation follows from elementary properties of characteristic functions , so that the characteristic function of a normally distributed random variable with parameters and is.
This proof sketch demonstrates the interaction of different mathematical disciplines, here stochastics and classical function theory .
literature
- Eugene Lukacs : Characteristic functions. Griffin, London 1960. 2nd edition 1970, ISBN 0-852-64170-2 .