In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the Lévy’s continuity theorem, named after the French mathematician Paul Lévy

Paul Pierre Lévy

Paul Pierre Lévy was a Jewish French mathematician who was active especially in probability theory, introducing martingales and Lévy flights...

, connects convergence in distribution of the sequence of random variables with pointwise convergence

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

 of their characteristic functions

Characteristic function (probability theory)

In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...

. An alternative name sometimes used is Lévy’s convergence theorem.

This theorem is the basis for one approach to prove the central limit theorem

Central limit theorem

In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

 and it is one of the major theorems concerning characteristic functions.

Theorem

Suppose we have

• a sequence of random variable
  Random variable
  In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
  
  s , not necessarily sharing a common probability space
  Probability space
  In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
  
  ,
  
• the sequence of corresponding characteristic function
  Characteristic function (probability theory)
  In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...
  
  s , which by definition are

  

  
  where E is the expected value
  Expected value
  In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
  
   operator.
  

If the sequence of characteristic functions converges pointwise

Pointwise convergence

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

 to some function φ
then the following statements become equivalent:

• converges in distribution to some random variable
  Random variable
  In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
  
   X
  i.e. the cumulative distribution functions corresponding to random variables converge at every continuity point;
  
• is tight
  Tightness of measures
  In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."-Definitions:...
  
  :
  
• φ(t) is a characteristic function of some random variable X;
  
• φ(t) is a continuous function
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
   of t;
  
• φ(t) is continuous
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
  at t = 0.
  

External links

• Lecture notes of "Theory of Probability" from MIT Open Course Sessions 9–14 are related to this theorem.