Definitions

# Lévy continuity theorem

The Lévy continuity theorem in probability theory, named after the French mathematician Paul Lévy, is the basis for one approach to prove the central limit theorem and it is one of the central theorems concerning characteristic functions.

Suppose we have

• a sequence of random variables $scriptstyle \left(X_n\right)_\left\{n=1\right\}^infty$ not necessarily sharing a common probability space, and
• the corresponding sequence of characteristic functions $scriptstyle \left(varphi_n\right)_\left\{n=1\right\}^infty$, which by definition are
• : $varphi_n\left(t\right)=E!left\left(e^\left\{itX_n\right\} right\right)quadforall tinmathbb\left\{R\right\}, quadforall ninmathbb\left\{N\right\}$

(where $E$ is the expected value operator). The theorem states that if the sequence of characteristic functions converge pointwise to a function $scriptstyle varphi$, i.e.

$forall tinmathbb\left\{R\right\} : varphi_n\left(t\right)tovarphi\left(t\right)$
then the following statements become equivalent,

• $scriptstyle X_n$ converges in distribution to some random variable $scriptstyle X$

$X_n xrightarrow\left\{mathcal D\right\} X$ i.e. the cumulative distribution functions corresponding to random variables converge(see convergence in distribution)

• $scriptstyle \left(X_n\right)_\left\{n=1\right\}^infty$ is tight, i.e.

$lim_\left\{xtoinfty\right\}left\left(sup_n P\left(|X_n|>x \right)right\right) = 0$

• $scriptstyle varphi\left(t\right)$ is a characteristic function of some random variable $scriptstyle X.$
• $scriptstyle varphi\left(t\right)$ is a continuous function of $scriptstyle t$.
• $scriptstyle varphi\left(t\right)$ is continuous at $scriptstyle t=0$.

An immediate corollary that is useful in proving the central limit theorem is that, $scriptstyle \left(X_n\right)_\left\{n=1\right\}^infty$ converges in distribution to some random variable $scriptstyle X$ with the characteristic function $scriptstyle varphi$ if it is the pointwise convergent limit of $scriptstyle \left(varphi_n\right)_\left\{n=1\right\}^infty$ and $scriptstyle varphi\left(t\right)$ is continuous at $scriptstyle t=0$.

### Proof

Rigorous proof of this theorem is available in A modern approach to probability theory by Bert Fristedt and Lawrence Gray (1997): Theorem 18.21