Characteristic functions
Section of Expectation and convergence, part of Stochastic processes
Characteristic functions
| Type | Section |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
The characteristic function \varphi_X(t)=\mathbb{E} e^{itX} exists for every real t, because |e^{itX}|=1. It determines the distribution. For independent random variables the characteristic function of a sum is the product of the characteristic functions, the same formal convenience that moment generating functions give on the set where they exist. The characteristic function exists without that restriction.
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
Lévy’s continuity theorem · Monotone convergence, Fatou, and dominated convergence · Modes of convergence · Strong law · Uniform integrability · Inequalities and limits under the expectation · Four modes of convergence · Where the reverse implications fail