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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.
This will obviously be also bounded and continuous, and therefore by the portmanteau lemma for sequence {X n} converging in distribution to X, we will have that E[g(X n)] → E[g(X)]. However the latter expression is equivalent to “E[ f ( X n , c )] → E[ f ( X , c )]”, and therefore we now know that ( X n , c ) converges in distribution ...
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 reason for the name is that if is an event in , then the theorem says that [] almost surely, i.e., the limit of the probabilities is 0 or 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be.
The dominated convergence theorem applies also to measurable functions with values in a Banach space, with the dominating function still being non-negative and integrable as above. The assumption of convergence almost everywhere can be weakened to require only convergence in measure.
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 ...
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