In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, Lévy’s continuity theorem, or Lévy's convergence theorem,
named after the French
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Paul Lévy, connects
convergence in distribution
In probability theory, there exist several different notions of convergence of sequences of random variables, including ''convergence in probability'', ''convergence in distribution'', and ''almost sure convergence''. The different notions of conve ...
of the sequence of random variables with
pointwise convergence
In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
of their
characteristic functions.
This theorem is the basis for one approach to prove the
central limit theorem
In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the Probability distribution, distribution of a normalized version of the sample mean converges to a Normal distribution#Standard normal distributi ...
and is one of the major theorems concerning characteristic functions.
Statement
Suppose we have
If the sequence of characteristic functions
converges pointwise to some function
:
then the following statements become equivalent:
Proof
Rigorous proofs of this theorem are available.
References
{{DEFAULTSORT:Levy continuity theorem
Theorems in probability theory
Theorems in statistics
Paul Lévy (mathematician)