Almost sure convergence is a concept in probability theory that describes the behavior of a sequence of random variables, where the sequence converges to a limit with probability one as the number of observations approaches infinity. This means that for almost all outcomes, the values of the random variables will get arbitrarily close to the limit eventually and stay close as more observations are made. It is a stronger form of convergence compared to convergence in probability and is closely related to the law of large numbers.
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