Abstract
In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values of the bits of X. In the classical model, the martingales considered are real-valued, that is, the bets made by the martingale can be arbitrary real numbers. In this paper, we investigate a more restricted model, where only integer-valued martingales are considered, and we study the class of random sequences induced by this model.
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References
Bienvenu, L., Merkle, W.: Constructive equivalence relations for computable probability measures. Annals of Pure and Applied Logic 160, 238–254 (2009)
Billingsley, P.: Probability and measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York (1995); A Wiley-Interscience Publication
Downey, R., Griffiths, E., Reid, S.: On Kurtz randomness. Theoretical Computer Science 321(2-3), 249–270
Doob, J.L.: Stochastic Processes. John Wiley & Sons Inc., New York (1953)
Hoeffding, W.: Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association 58, 13–30 (1963)
Kakutani, S.: On equivalence of infinite product measures. Annals of Mathematics 49(214-224) (1948)
Kurtz, S.: Randomness and genericity in the degrees of unsolvability. PhD dissertation, University of Illinois at Urbana (1981)
Merkle, W., Miller, J.S., Nies, A., Reimann, J., Stephan, F.: Kolmogorov-Loveland randomness and stochasticity. Annals of Pure and Applied Logic 138(1-3), 183–210 (2006)
Muchnik, A.A., Semenov, A., Uspensky, V.: Mathematical metaphysics of randomness. Theoretical Computer Science 207(2), 263–317 (1998)
Nies, A.: Computability and Randomness. Oxford Logic Guides, vol. 51. Oxford University Press, Oxford (2009)
Ross, S.M.: Stochastic Processes, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York (1996)
Schnorr, C.-P.: Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie. LNM, vol. 218. Springer, Berlin (1971)
Shen, A.: On relations between different algorithmic definitions of randomness. Soviet Mathematics Doklady 38, 316–319 (1989)
Vovk, V.: On a criterion for randomness. Soviet Mathematics Doklady 294(6), 1298–1302 (1987)
Wang, Y.: A separation of two randomness concepts. Information Processing Letters 69(3), 115–118 (1999)
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Bienvenu, L., Stephan, F., Teutsch, J. (2010). How Powerful Are Integer-Valued Martingales?. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_7
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DOI: https://doi.org/10.1007/978-3-642-13962-8_7
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