Darmois-Skitowitsch theorem

The set of Darmois-Skitovich is a sentence from the stochastics , the normal distribution through independent linear forms characterized by random variables. The theorem is important in mathematical statistics , since the distribution is usually not known there.

It is named after the French mathematician Georges Darmois and the Russian mathematician Viktor Pawlowitsch Skitowitsch .

Formulation of the sentence

Be independent random variables and . Are now and independent, then all are normally distributed. ${\displaystyle (X_{i})_{i=1}^{n}}$ ${\displaystyle \alpha _{i},\beta _{i}\neq 0,\forall i=1,\dots ,n}$ ${\displaystyle L_{1}=\sum _{i=1}^{n}\alpha _{i}X_{i}}$ ${\displaystyle L_{2}=\sum _{i=1}^{n}\beta _{i}X_{i}}$ ${\displaystyle (X_{i})_{i=1}^{n}}$

Explanation of the sentence

The theorem shows that the independence of the linear forms of the random variables is sufficient to characterize the normal distribution and dispenses with the condition of the identical distribution .

Individual evidence

1. ^ Mathematical Statistics , p. 97, by Ludger Rüschendorf, 1970. Google books excerpt