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When X n converges almost completely towards X then it also converges almost surely to X. In other words, if X n converges in probability to X sufficiently quickly (i.e. the above sequence of tail probabilities is summable for all ε > 0), then X n also converges almost surely to X. This is a direct implication from the Borel–Cantelli lemma.
If X n are independent random variables assuming value one with probability 1/n and zero otherwise, then X n converges to zero in probability but not almost surely. This can be verified using the Borel–Cantelli lemmas.
In probability theory, Kolmogorov's Three-Series Theorem, named after Andrey Kolmogorov, gives a criterion for the almost sure convergence of an infinite series of random variables in terms of the convergence of three different series involving properties of their probability distributions.
Convergence of random variables, for "almost sure convergence" With high probability; Cromwell's rule, which says that probabilities should almost never be set as zero or one; Degenerate distribution, for "almost surely constant" Infinite monkey theorem, a theorem using the aforementioned terms; List of mathematical jargon
The proof can also be based on Fatou's lemma instead of a direct proof as above, because Fatou's lemma can be proved independent of the monotone convergence theorem. However the monotone convergence theorem is in some ways more primitive than Fatou's lemma. It easily follows from the monotone convergence theorem and proof of Fatou's lemma is ...
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any L p space. In order to obtain convergence in L 1 (i.e., convergence in mean), one requires uniform integrability of the random variables .
In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers .
Using the definition of X, its representation as pointwise limit of the Y k, the monotone convergence theorem for conditional expectations, the last inequality, and the definition of the limit inferior, it follows that almost surely