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. ${\displaystyle X}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$ ${\displaystyle X_{1}}$ ${\displaystyle X_{2}}$

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. ${\displaystyle X_{1}+X_{2}}$

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. ${\displaystyle X=X_{1}+X_{2}}$ ${\displaystyle \varphi (t)=\varphi _{1}(t)\cdot \varphi _{2}(t)}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi _{1},\,\varphi _{2}}$ ${\displaystyle \varphi _{1}(t)=\exp(a_{0}+a_{1}\cdot t+a_{2}\cdot t^{2})}$ ${\displaystyle \varphi _{1}(t)=\exp(\mathrm {i} a_{1}t-a_{2}t^{2})}$ ${\displaystyle \varphi _{1}}$ ${\displaystyle \mu =a_{1}\,}$ ${\displaystyle \sigma ^{2}=2\cdot a_{2}}$

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 .

Individual evidence

1. ^ Paul Lévy: Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées. In: J. Math. Pures Appl. 14, 1935, pp. 347-402.
2. ↑ Harald Cramér: About a property of the normal distribution function. In: Math. Z. 41, 1936, pp. 405-414.