Inequalities and limits under the expectation
Section of Expectation and convergence, part of Stochastic processes
Inequalities and limits under the expectation
| Type | Section |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
Expectation is linear on integrable variables, and it is monotone: X\le Y implies \mathbb{E} X\le\mathbb{E} Y when both expectations exist. If \varphi is convex and X is integrable, with \varphi(X) integrable as well, Jensen’s inequality says
\varphi(\mathbb{E} X)\le\mathbb{E}\varphi(X).
For \varphi(x)=x^2 this is (\mathbb{E} X)^2\le\mathbb{E} X^2, which rearranges to \operatorname{Var}(X)\ge 0.
Theorem 1 (Monotone convergence, Fatou, and dominated convergence) 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.
Article: Monotone convergence, Fatou, and dominated convergence →
The hypotheses differ. An integrable bound that dominates every X_n stops probability mass from escaping into spikes that get rarer and taller. Without that bound, a limit of the variables need not be the limit of the expectations.
Example 1 (A spike of height n and width 1/n) Let U be uniform on (0,1) and let X_n=n\,1_{\{U\le 1/n\}}.
Question. Find the almost-sure limit and the expectations \mathbb{E} X_n.
Solution. If U(\omega)=u>0, then u>1/n for every n>1/u, so X_n(\omega)=0 from that index on. The single point U=0 has probability 0. Thus X_n\to 0 with probability 1. The expectation does not follow:
\mathbb{E} X_n=n\cdot\mathbb{P}(U\le 1/n)=n\cdot\frac{1}{n}=1
for every n. There is no integrable Y with |X_n|\le Y for all n. Any such Y would have to be at least n on a set of probability 1/n, for every n, and those spikes force \mathbb{E} Y=\infty.
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See also
Monotone convergence, Fatou, and dominated convergence · Modes of convergence · Strong law · Uniform integrability · Four modes of convergence · Where the reverse implications fail · Laws of large numbers · Characteristic functions