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 Teor. Veroyatnost. i Primenen., 1999, Volume 44, Issue 3, Pages 617–630 (Mi tvp806)

Purely game-theoretic random sequences: I. Strong law of large numbers and law of the iterated logarithm

M. Minozzo

Department of Statistical Sciences, University of Perugia, Italy

Abstract: Random sequences are usually defined with respect to a probability distribution $\mathbf{P}$ (a $\sigma$-additive set function, normed to one, defined over a $\sigma$-algebra) assuming Kolmogorov's axioms for probability theory. In this paper, without using this axiomatics, we give a definition of random (typical) sequences taking as primitive the notion of a martingale and using the principle of the excluded gambling strategy. In this purely game-theoretic framework, no probability distribution or, partially or fully specified, system of conditional probability distributions needs to be introduced. For these typical sequences, we prove direct algorithmic versions of Kolmogorov's strong law of large numbers (SLLN) and of the upper half of Kolmogorov's law of the iterated logarithm (LIL).

Keywords: algorithmic probability theory, almost sure limit theorems, martingales, typical sequences.

DOI: https://doi.org/10.4213/tvp806

Full text: PDF file (804 kB)

English version:
Theory of Probability and its Applications, 2000, 44:3, 511–522

Bibliographic databases:

Revised: 11.11.1998
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Citation: M. Minozzo, “Purely game-theoretic random sequences: I. Strong law of large numbers and law of the iterated logarithm”, Teor. Veroyatnost. i Primenen., 44:3 (1999), 617–630; Theory Probab. Appl., 44:3 (2000), 511–522

Citation in format AMSBIB
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\paper Purely game-theoretic random sequences:~I. Strong law of large numbers and law of the iterated logarithm
\jour Teor. Veroyatnost. i Primenen.
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\pages 617--630
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\jour Theory Probab. Appl.
\yr 2000
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\pages 511--522
\crossref{https://doi.org/10.1137/S0040585X97977768}