Betting strategies with bounded splits
Abstract
We show that a pair of Kolmogorov-Loveland betting strategies cannot win on every non-Martin-L\"of 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 , bet on at most -g() positions among the first 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 , bet on at least -g() positions among the first positions of the sequence.
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