Monotone convergence, Fatou, and dominated convergence
Theorem taught in Expectation and convergence, part of Stochastic processes
Monotone convergence, Fatou, and dominated convergence
| Type | Theorem |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
If 0\le X_n\uparrow X with probability 1, the monotone convergence theorem gives \mathbb{E} X_n\uparrow\mathbb{E} X, and the limit may be infinite. If X_n\ge 0, Fatou’s lemma gives \mathbb{E}(\liminf_n X_n)\le\liminf_n\mathbb{E} X_n. If X_n\to X with probability 1 and |X_n|\le Y for every n, with \mathbb{E} Y<\infty, the dominated convergence theorem gives \mathbb{E} X_n\to\mathbb{E} X.
Used in
- Taught in Expectation and convergence, where the derivation and examples are.
- All 3 questions on Expectation and convergence