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

12/27/2022

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by   Kohtaro Tadaki, et al.

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The law of large numbers is one of the fundamental properties which algorithmically random infinite sequences ought to satisfy. In this paper, we show that the law of large numbers can be effectivized for an arbitrary Schnorr random infinite sequence, with respect to an arbitrary computable Bernoulli measure. Moreover, we show that an absolute speed limit of convergence exists in this effectivization, and it equals 2 in a certain sense. In the paper, we also provide the corresponding effectivization of almost sure convergence in the strong law of large numbers, and its absolute speed limit of convergence, in the context of probability theory, with respect to a large class of probability spaces and i.i.d. random variables on them, which are not necessarily computable.