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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 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" Infinite monkey theorem, a theorem using the aforementioned terms; List of mathematical jargon
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. This can be verified using the Borel–Cantelli lemmas.
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 () = ().
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 .
On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {X n}. The second term converges to zero as δ → 0, since the set B δ shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem.
The limiting distribution of the sequence is a degenerate random variable which equals θ 0 with probability 1. In statistics , a consistent estimator or asymptotically consistent estimator is an estimator —a rule for computing estimates of a parameter θ 0 —having the property that as the number of data points used increases indefinitely ...