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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 of random variables such that the almost surely convergent sequences are exactly the converging sequences with respect to that topology. In particular, there is no metric of almost sure convergence.
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, an event is said to happen almost surely ... the probability converges to 1. This is equivalent to convergence in probability.
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 () = ().
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).
Then the sequence converges almost surely to a random variable with finite expectation. There is a symmetric statement for submartingales with bounded expectation of the positive part. A supermartingale is a stochastic analogue of a non-increasing sequence, and the condition of the theorem is analogous to the condition in the monotone ...
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 .
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.