# Characterization theorems

in probability theory and mathematical statistics

Theorems that establish a connection between the type of the distribution of random variables or random vectors and certain general properties of functions in them.

## Contents

### Example 1.

Let $X$ be a three-dimensional random vector such that:

1) its projections $X _ {1} , X _ {2} , X _ {3}$ onto any three mutually-orthogonal axes are independent; and

2) the density $p ( x)$, $x = ( x _ {1} , x _ {2} , x _ {3} )$, of the probability distribution of $X$ depends only on $x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2}$. Then the distribution of $X$ is normal and

$$p ( x) = \ \frac{1}{( 2 \pi ) ^ {3/2} \sigma ^ {2} } \ \mathop{\rm exp} \left \{ - \frac{1}{2 \sigma ^ {2} } ( x _ {1} ^ {2} + x _ {2} ^ {2} + x _ {3} ^ {2} ) \right \} ,$$

where $\sigma > 0$ is a certain constant (the Maxwell law for the distribution of the velocities of molecules in a gas in stationary state).

### Example 2.

Let $X \in \mathbf R ^ {n}$ be a random vector with independent and identically-distributed components $X = ( X _ {1} \dots X _ {n} )$. If the distribution is normal then the "sample meansample mean"

$$\overline{X}\; = \ { \frac{1}{n} } \sum _ {j = 1 } ^ { n } X _ {j}$$

and the "sample variancesample variance"

$$\overline{ {s ^ {2} }}\; = \ { \frac{1}{n} } \sum _ {j = 1 } ^ { n } ( X _ {j} - \overline{X}\; ) ^ {2}$$

are independent random variables. Conversely, if they are independent, then the distribution of $X$ is normal.

### Example 3.

Let $X \in \mathbf R ^ {n}$ be a vector with independent and identically-distributed components. There are non-zero constants $a _ {1} \dots a _ {n}$, $b _ {1} \dots b _ {n}$ such that the random variables

$$Y _ {1} = \ a _ {1} X _ {1} + \dots + a _ {n} X _ {n}$$

and

$$Y _ {2} = \ b _ {1} X _ {1} + \dots + b _ {n} X _ {n}$$

are independent if and only if $X$ has a normal distribution. The last assertion remains true if the assumption that $Y _ {1}$ and $Y _ {2}$ are independent is replaced by the assumption that they are identically distributed, adding, however, certain restrictions on the coefficients $a _ {j}$ and $b _ {j}$.

A characterization of a similar kind of the distribution of a random vector $X \in \mathbf R ^ {n}$ by the property of identical distribution or of independence of two polynomials $Q _ {1} ( X)$ and $Q _ {2} ( X)$ is given by a number of characterization theorems that play an important role in mathematical statistics.

How to Cite This Entry:
Characterization theorems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Characterization_theorems&oldid=46324
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article