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Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. It is the notion of convergence used in the strong law of large numbers. The concept of almost sure convergence does not come from a topology on the space of random variables. This means there is no topology on the space ...
Almost everywhere, the corresponding concept in measure theory; 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"
Convergence in probability does not imply almost sure convergence in the discrete case [ edit ] If X n are independent random variables assuming value one with probability 1/ n and zero otherwise, then X n converges to zero in probability but not almost surely.
In probability theory, the continuous mapping theorem states that continuous functions preserve limits even if their arguments are sequences of random variables. A continuous function, in Heine's definition, is such a function that maps convergent sequences into convergent sequences: if x n → x then g(x n) → g(x).
Probability theory or probability calculus is the branch of mathematics concerned with probability.Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms.
Note that almost uniform convergence of a sequence does not mean that the sequence converges uniformly almost everywhere as might be inferred from the name. However, Egorov's theorem does guarantee that on a finite measure space, a sequence of functions that converges almost everywhere also converges almost uniformly on the same set.
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.
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 .