How Powerful Are Integer-Valued Martingales?

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.

[1]  Denis R. Hirschfeldt,et al.  Algorithmic randomness and complexity. Theory and Applications of Computability , 2012 .

[2]  ComplexityValentine KabanetsDecember.  Randomness and Complexity , 1997 .

[3]  Jason Teutsch,et al.  How to build a probability-free casino , 2012, Inf. Comput..

[4]  C. Schnorr.  Zufälligkeit und Wahrscheinlichkeit , 1971 .

[5]  J. Doob.  Stochastic processes , 1953 .

[6]  Stephanie Reid,et al.  On Kurtz randomness , 2004, Theor. Comput. Sci..

[7]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

[8]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[9]  Jack H. Lutz,et al.  Why Computational Complexity Requires Stricter Martingales , 2002, ICALP.

[10]  Rodney G. Downey,et al.  Algorithmic Randomness and Complexity , 2010, Theory and Applications of Computability.

[11]  Kazuoki Azuma.  WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .

[12]  Yongge Wang.  A Separation of Two Randomness Concepts , 1999, Inf. Process. Lett..

[13]  Claus-Peter Schnorr,et al.  Zufälligkeit und Wahrscheinlichkeit - Eine algorithmische Begründung der Wahrscheinlichkeitstheorie , 1971, Lecture Notes in Mathematics.

[14]  Andrej Muchnik,et al.  Mathematical Metaphysics of Randomness , 1998, Theor. Comput. Sci..

[15]  Claus-Peter Schnorr,et al.  A unified approach to the definition of random sequences , 1971, Mathematical systems theory.

[16]  Robert I. Soare,et al.  Recursively enumerable sets and degrees - a study of computable functions and computability generated sets , 1987, Perspectives in mathematical logic.

[17]  Jack H. Lutz,et al.  Why Computational Complexity Requires Stricter Martingales , 2005, Theory of Computing Systems.

[18]  André Nies,et al.  Kolmogorov-Loveland randomness and stochasticity , 2005, Ann. Pure Appl. Log..

[19]  S. Kakutani.  On Equivalence of Infinite Product Measures , 1948 .

[20]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[21]  Wolfgang Merkle,et al.  Constructive equivalence relations on computable probability measures , 2006, Ann. Pure Appl. Log..

[22]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..