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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.
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.
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.
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).
Law 3 is called the strong law because random variables which converge strongly (almost surely) are guaranteed to converge weakly (in probability). However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak (in probability).
When successive powers of a matrix T become small (that is, when all of the entries of T approach zero, upon raising T to successive powers), the matrix T converges to the zero matrix. A regular splitting of a non-singular matrix A results in a convergent matrix T. A semi-convergent splitting of a matrix A results in a semi-convergent matrix T.
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any L p space. In order to obtain convergence in L 1 (i.e., convergence in mean), one requires uniform integrability of the random variables .