In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc.
More precisely, the model of characterization of probability distribution was described by V.M.
of random variables with values in measurable metric space
of random variables with values in measurable metric space
By characterizations of probability distributions we understand general problems of description of some set
which describe the properties of random variables
, obtained by means of a specially chosen mapping
The description of the properties of the random variables
So, the set which interests us appears therefore in the following form: where
denotes the complete inverse image of
This is the general model of characterization of probability distribution.
Verification of conditions of characterization theorems in practice is possible only with some error
[5] Such a situation is observed, for instance, in the cases where a sample of finite size is considered.
Suppose that the conditions of the characterization theorem are fulfilled not exactly but only approximately.
May we assert that the conclusion of the theorem is also fulfilled approximately?
The theorems in which the problems of this kind are considered are called stability characterizations of probability distributions.