Lévy's Continuity Theorem

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:

$\ds \lim_{M \mathop \to \infty} \sup_{n \mathop \ge 1} P \sqbrk {\size {X_n} \ge M} = 0$

Proof

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Source of Name

This entry was named for Paul Pierre Lévy.

Sources

This needs considerable tedious hard slog to complete it. In particular: Please indicate, by means of chapter / section / equation reference (or however it is indicated) where in the book it can be found. Not page name, for obvious reasons. Second edition is presumed. There is no 2018 edition, as far as I can tell. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

• 2002: R.M. Dudley: Real Analysis and Probability (2nd ed.)