Lévy’s continuity theorem
Topic of Expectation and convergence, part of Stochastic processes
Lévy’s continuity theorem
| Type | Topic |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
If the characteristic functions \varphi_{X_n}(t) converge for every t to a function \varphi that is continuous at 0, then X_n converges in distribution to the law with characteristic function \varphi. It is one route to the central limit theorem.
In Expectation and convergence
Lévy’s continuity theorem says that if \varphi_{X_n}(t)\to\varphi(t) for every t, and \varphi is continuous at 0 and is itself a characteristic function, then X_n converges in distribution to the law with characteristic function \varphi. This is one route to the central limit theorem: the characteristic function of the standardised sum converges to e^{-t^2/2}. Convergence of characteristic functions does not, by itself, give convergence of moments.
Used in
See also
Characteristic functions · Modes of convergence · Central limit theorem · Monotone convergence, Fatou, and dominated convergence · Strong law · Uniform integrability · Inequalities and limits under the expectation · Four modes of convergence