Search results
Results From The WOW.Com Content Network
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 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
If is a stationary ergodic process, then () converges almost surely to = [] . The Glivenko–Cantelli theorem gives a stronger mode of convergence than this in the iid case. An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm .
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 () = ().
This expression asserts the pointwise convergence of the empirical distribution function to the true cumulative distribution function. There is a stronger result, called the Glivenko–Cantelli theorem, which states that the convergence in fact happens uniformly over t: [5]
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...