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The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
Let in the theorem denote a random variable that takes the values / and / with equal probabilities. With = the summands of the first two series are identically zero and var(Y n)=. The conditions of the theorem are then satisfied, so it follows that the harmonic series with random signs converges almost surely.
This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {X n} converges in distribution to X if and only if any of the following conditions are met:
In the theory of probability, the Glivenko–Cantelli theorem (sometimes referred to as the Fundamental Theorem of Statistics), named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, describes the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. [1]
In probability theory, Slutsky's theorem extends some properties of algebraic operations on convergent sequences of real numbers to sequences of random variables. [1] The theorem was named after Eugen Slutsky. [2] Slutsky's theorem is also attributed to Harald Cramér. [3]
Download as PDF; Printable version; From Wikipedia, the free encyclopedia. Redirect page. Redirect to: Convergence of random variables#Convergence in distribution;
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]
Uniform convergence in probability has applications to statistics as well as machine learning as part of statistical learning theory. The law of large numbers says that, for each single event A {\displaystyle A} , its empirical frequency in a sequence of independent trials converges (with high probability) to its theoretical probability.