In
probability theory
Probability theory 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 expressing it through a set o ...
, 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, structure, space, models, and change.
History
On ...
Paul Lévy, connects
convergence in distribution
In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
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 functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
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) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
and it 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
Probability theorems
Theorems in statistics
Paul Lévy (mathematician)