Lévy's continuity theorem

In probability theory, Lévy’s continuity theorem, named after the French mathematician Paul Lévy, connects convergence in distribution of the sequence of random variables with pointwise convergence of their characteristic functions. An alternative name sometimes used is Lévy’s convergence theorem.^[1]

This theorem is the basis for one approach to prove the central limit theorem and it is one of the major theorems concerning characteristic functions.

Theorem

Suppose we have

a sequence of random variables $\scriptstyle \{X_n\}_{n=1}^\infty$ , not necessarily sharing a common probability space,
the sequence of corresponding characteristic functions $\scriptstyle \{\varphi_n\}_{n=1}^\infty$ , which by definition are
$\varphi_n(t) = \operatorname{E}\, e^{itX_n} \quad \forall t\in\mathbb{R},\ \forall n\in\mathbb{N},$
where $\operatorname {E}$ is the expected value operator.

If the sequence of characteristic functions converges pointwise to some function $\varphi$

\varphi_n(t)\to\varphi(t) \quad \forall t\in\mathbb{R},

then the following statements become equivalent:

$X_{n}$ converges in distribution to some random variable X
$X_n\ \xrightarrow{\mathcal D}\ X,$
i.e. the cumulative distribution functions corresponding to random variables converge at every continuity point;
$\scriptstyle \{X_n\}_{n=1}^\infty$ is tight:
$\lim_{x\to\infty}\left( \sup_n \operatorname{P}\big[\, |X_n|>x \,\big]\right) = 0;$
$\varphi (t)$ is a characteristic function of some random variable X;
$\varphi (t)$ is a continuous function of t;
$\varphi (t)$ is continuous at t = 0.

Proof

Rigorous proofs of this theorem are available.^[1]^[2]

Notes

1 2 Williams (1991, section 18.1)
↑ Fristedt & Gray (1996, Theorem 18.21)

References

Williams, D. (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6.
Fristedt, B. E.; Gray, L. F. (1996): A modern approach to probability theory, Birkhäuser Boston. ISBN 0-8176-3807-5

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