Lévy's Continuity Theorem

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Theorem

Let $\sequence {X_n}_{n \mathop \ge 1}$ be a sequence of discrete random variables with characteristic functions $\map {\phi_n} t := E \sqbrk {e^{i t X_n} }$.


Let the sequence $\sequence {\phi_n}$ converge to some real function $\phi$:

$\forall t \in \R: \map {\phi_n} t \to \map \phi t$.


Then the following statements are equivalent:

$(1): \quad$ The $\sequence {X_n}$ converges in distribution to some random variable $X$:
$X_n \stackrel {dist} {\to} X$ with characteristic function $\map {\phi_X} t := \map \phi t$


$(2): \quad$ The sequence $\sequence {X_n}$ is tight, that is:
$\displaystyle \lim_{M \mathop \to \infty} \sup_{n \mathop \ge 1} P \sqbrk {\size {X_n} \ge M} = 0$


Proof


Source of Name

This entry was named for Paul Pierre Lévy.


Sources