Asymptotic Divergences and Strong Dichotomy
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The Schnorr-Stimm dichotomy theorem concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet Σ. In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergencediv(S||α) of a probability measure α on Σ from a sequence S over Σ and the upper asymptotic divergenceDiv(S||α) of α from S in such a way that a sequence S is α-normal (meaning that every string w has asymptotic frequency α(w) in S) if and only if Div(S||α)=0. We also use the Kullback-Leibler divergence to quantify the total riskRisk_G(w) that a finite-state gambler G takes when betting along a prefix w of S. Our main theorem is a strong dichotomy theorem that uses the above notions to quantify the exponential rates of winning and losing on the two sides of the Schnorr-Stimm dichotomy theorem (with the latter routinely extended from normality to α-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S. (1) The infinitely-often exponential rate of winning is 2^Div(S||α)|w|. (2) The exponential rate of loss is 2^-Risk_G(w). We also use (1) to show that 1-Div(S||α)/c, where c= log(1/ min_a∈Σα(a)), is an upper bound on the finite-state α-dimension of S and prove the dual fact that 1-div(S||α)/c is an upper bound on the finite-state strong α-dimension of S.
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