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.

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.

References

[1]	A.M. Kagan, Yu.V. Linnik, S.R. Rao, "Characterization problems in mathematical statistics" , Wiley (1973) (Translated from Russian)