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In probability theory, there exist several different notions of convergence of sequences of random variables, including convergence in probability, convergence in distribution, and almost sure convergence. The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than ...
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 experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between almost surely and surely (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, [2] because an infinite set can ...
The order in probability notation is used in probability theory and statistical theory in direct parallel to the big O notation that is standard in mathematics.Where the big O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in ...
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 .
Doob's martingale convergence theorems imply that conditional expectations also have a convergence property. Let ( Ω , F , P ) {\displaystyle (\Omega ,F,\mathbf {P} )} be a probability space and let X {\displaystyle X} be a random variable in L 1 {\displaystyle L^{1}} .
The problem of uniform convergence of P n to P was open until Vapnik and Chervonenkis solved it in 1968. [1] If the class (or ) is Glivenko–Cantelli with respect to P then P n converges to P uniformly over (or ). In other words, with probability 1 we have
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.