Lévy continuity theorem

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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 $\scriptstyle (X_n)_{n=1}^\infty$ not necessarily sharing a common probability space, and
the corresponding sequence of characteristic functions $\scriptstyle (\varphi_n)_{n=1}^\infty$ , which by definition are

$\varphi_n(t)=E\!\left(e^{itX_n} \right)\quad\forall t\in\mathbb{R}, \quad\forall n\in\mathbb{N}$

(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.

$\forall t\in\mathbb{R} : \varphi_n(t)\to\varphi(t)$

then the following statements become equivalent,

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

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

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

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

$\scriptstyle \varphi(t)$ is a characteristic function of some random variable $\scriptstyle X.$

$\scriptstyle \varphi(t)$ is a continuous function of $\scriptstyle t$ .

$\scriptstyle \varphi(t)$ is continuous at $\scriptstyle t=0$ .

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