How powerful are integer-valued martingales?

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. 1 Gambling with or without coins One of the main approaches to define the notion of random sequence is the so-called “unpredictability paradigm. ” We say that an infinite binary sequence is “random ” if there is no effective way to win arbitrarily large amounts of money by betting on the values of its bits. The main notion arising from this paradigm is computable randomness, but other central notions such as Martin-Löf random