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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 ...
The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely. On the other hand, the analogous series of (for example) square root reciprocals with random signs, namely =,
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 .
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 .
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 ...
This theorem follows from the fact that if X n converges in distribution to X and Y n converges in probability to a constant c, then the joint vector (X n, Y n) converges in distribution to (X, c) . Next we apply the continuous mapping theorem , recognizing the functions g ( x , y ) = x + y , g ( x , y ) = xy , and g ( x , y ) = x y −1 are ...