Where the reverse implications fail
Section of Expectation and convergence, part of Stochastic processes
Where the reverse implications fail
| Type | Section |
| Taught in | Expectation and convergence |
| Part | Stochastic processes |
The spike X_n=n\,1_{\{U\le 1/n\}} converges almost surely to 0, and \mathbb{E}|X_n|=1 does not tend to 0. Almost-sure convergence does not imply L^1 convergence.
L^1 convergence does not imply almost-sure convergence. Enumerate the dyadic intervals of (0,1] level by level: at level k=0,1,2,\ldots the intervals are ((j-1)2^{-k},\, j 2^{-k}] for j=1,\ldots,2^k. Let X_n be the indicator of the nth interval in this list. The length of the interval at level k is 2^{-k}, which tends to 0 as n runs through the later levels, so \mathbb{E}|X_n|\to 0. Every point of (0,1] lies in exactly one interval at each level, so X_n(\omega)=1 for infinitely many n. The indicators do not converge to 0 at any \omega.
L^1 controls the average size of the error at each n. Almost-sure convergence asks whether one path eventually settles. A summable series of tail probabilities joins the two: if \sum_n\mathrm{P}(|X_n-X|>\varepsilon)<\infty for every \varepsilon>0, the first Borel–Cantelli lemma gives almost-sure convergence.
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See also
Monotone convergence, Fatou, and dominated convergence · Modes of convergence · Strong law · Uniform integrability · Inequalities and limits under the expectation · Four modes of convergence · Laws of large numbers · Characteristic functions