Modes of convergence
Definition taught in Expectation and convergence, part of Stochastic processes
Modes of convergence
| Type | Definition |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
- X_n\to X almost surely when \mathrm{P}\{\omega: X_n(\omega)\to X(\omega)\}=1.
- X_n\to X in probability when \mathrm{P}(|X_n-X|>\varepsilon)\to 0 for every \varepsilon>0.
- X_n\to X in L^p, for p\ge 1, when \mathbb{E}|X_n-X|^p\to 0.
- X_n\to X in distribution when F_{X_n}(x)\to F_X(x) at every continuity point x of F_X.
Used in
- Taught in Expectation and convergence, where the derivation and examples are.
- All 3 questions on Expectation and convergence
See also
Monotone convergence, Fatou, and dominated convergence · Strong law · Uniform integrability