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 the central theorems concerning characteristic functions.

Suppose we have

• a sequence of random variables $\backslash scriptstyle\; (X\_n)\_\{n=1\}^\backslash infty$ not necessarily sharing a common probability space, and
• the corresponding sequence of characteristic functions $\backslash scriptstyle\; (\backslash varphi\_n)\_\{n=1\}^\backslash infty$, which by definition are

  $\backslash varphi\_n(t)=E\backslash !\backslash left(e^\{itX\_n\}\; \backslash right)\backslash quad\backslash forall\; t\backslash in\backslash mathbb\{R\},\; \backslash quad\backslash forall\; n\backslash in\backslash mathbb\{N\}$

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

$\backslash forall\; t\backslash in\backslash mathbb\{R\} :\; \backslash varphi\_n(t)\backslash to\backslash varphi(t)$

then the following statements become equivalent,

• $\backslash scriptstyle\; X\_n$ converges in distribution to some random variable $\backslash scriptstyle\; X$

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

• $\backslash scriptstyle\; (X\_n)\_\{n=1\}^\backslash infty$ is tight, i.e.

$\backslash lim\_\{x\backslash to\backslash infty\}\backslash left(\; \backslash sup\_n\; P(\; |X\_n|>x\; )\backslash right)\; =\; 0$

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

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

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

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

Proof

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

External links

• Lecture Note of "Theory of Probability" from MIT Open Course Session 9-14 is related to this theorem.

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