Betting strategies with bounded splits

12/29/2022

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by   Tomislav Petrović, et al.

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We show that a pair of Kolmogorov-Loveland betting strategies cannot win on every non-Martin-Löf random sequence if either of the two following conditions is true: (I) There is an unbounded computable function g such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all ℓ, bet on at most ℓ-g(ℓ) positions among the first ℓ positions of the sequence. (II) There is a sublinear function g such that both betting strategies, when betting on an infinite binary sequence, almost surely, for almost all ℓ, bet on at least ℓ-g(ℓ) positions among the first ℓ positions of the sequence.