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The law of large numbers is one of the fundamental properties whichalgorithmically random infinite sequences ought to satisfy. In this paper, weshow that the law of large numbers can be effectivized for an arbitrary Schnorrrandom infinite sequence, with respect to an arbitrary computable Bernoullimeasure. Moreover, we show that an absolute speed limit of convergence existsin this effectivization, and it equals 2 in a certain sense. In the paper, wealso provide the corresponding effectivization of almost sure convergence inthe strong law of large numbers, and its absolute speed limit of convergence,in the context of probability theory, with respect to a large class ofprobability spaces and i.i.d. random variables on them, which are notnecessarily computable.

An effectivization of the law of large numbers for algorithmically random sequences and its absolute speed limit of convergence