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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 ...
It is equivalent to check condition (iii) for the series = = = (′) where for each , and ′ are IID—that is, to employ the assumption that [] =, since is a sequence of random variables bounded by 2, converging almost surely, and with () = ().
In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
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
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 .
For every (fixed) x, F n (x) is a sequence of random variables which converge to F(x) almost surely by the strong law of large numbers. That is, F n converges to F pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of F n to F by the Glivenko–Cantelli theorem. [2]
Since the sequence is uniformly bounded, there is a real number M such that |f n (x)| ≤ M for all x ∈ S and for all n. Define g(x) = M for all x ∈ S. Then the sequence is dominated by g. Furthermore, g is integrable since it is a constant function on a set of finite measure. Therefore, the result follows from the dominated convergence ...