Lévy continuity theorem

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The Lévy continuity theorem in probability theory is the basis for one approach to prove the central limit theorem.

Suppose we have

a sequence of random variables $[X n]$ NOT necessarily sharing a common probability space, and
the corresponding sequence of characteristic functions $[φ n (t)]$ where

$\phi_n(t):=\mathbb{E}[e^{itX_n}]\quad\forall t\in\mathbb{R}$

The theorem states that if the pointwise convergent $φ(t)$ exists for the sequence of characteristic functions, i.e.

$(\forall t\in\mathbb{R})(\phi_n(t)\to\phi(t))$

then the following statements become equivalent,

$X_n\to X$ in distribution for some random variable $X$ , i.e.

$\Pr[X_n\le x]\rightarrow\Pr[X\le x] \quad \forall x\in C_X:=\{x:\Pr[X\le x]\mbox{ is continuous}\}$
$[X n]$ is tight, i.e.

$\lim_{x\to\infty} \sup_n \Pr[ |X_n|>x ] = 0$
$φ(t)$ is a characteristic function of some random variable $X$
$φ(t)$ is a continuous function of $t$
$φ(t)$ is continuous at $t = 0$

An immediate corollary that is useful in proving the central limit theorem is that, $[X n]$ converges in distribution to some random variable $X$ all having the characteristic function $φ(t)$ if $φ(t)$ is the pointwise convergent continuous at $t = 0$ of $[φ n (t)]$ .