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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 ...
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, 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.
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). [1] In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty.
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]
Strong convergence is also known as almost sure convergence. Most common shorthand notation: ... In fact, strong convergence implies convergence in probability, and ...
Given a discrete-time stationary ergodic stochastic process on the probability space (,,), the asymptotic equipartition property is an assertion that, almost surely, (,, …,) where () or simply denotes the entropy rate of , which must exist for all discrete-time stationary processes including the ergodic ones.
In order to obtain convergence in L 1 (i.e., convergence in mean), one requires uniform integrability of the random variables . By Chebyshev's inequality, convergence in L 1 implies convergence in probability and convergence in distribution. The following are equivalent: