Abstract
The Schnorr-Stimm dichotomy theorem (Schnorr and Stimm, 1972) concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet Σ. The theorem asserts that, for any such sequence S, the following two things are true. (1) If S is not normal in the sense of Borel (meaning that every two strings of equal length appear with equal asymptotic frequency in S), then there is a finite-state gambler that wins money at an infinitely-often exponential rate betting on S. (2) If S is normal, then any finite-state gambler loses money at an exponential rate betting on S. In this paper we use the Kullback-Leibler divergence to formulate the lower asymptotic divergence div(S||α) of a probability measure α on Σ from a sequence S over Σ and the upper asymptotic divergence Div(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 risk Risk <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</sub> (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 in 1 is 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Div(S||α)|w|</sup> . ( $2~'$ ) The exponential rate of loss in 2 is 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">- Risk<sub>G</sub>(w)</sup> . We also use (1 $'$ ) to show that 1- Div(S||α)/c, where c = log(1/ min <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a ∈ Σ</sub> α(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.
Highlights
An infinite sequence S over a finite alphabet is normal in the 1909 sense of Borel [7] if every two strings of equal length appear with equal asymptotic frequency in S
Schnorr [30] gave an equivalent, and more flexible, formulation of Martin-Löf’s notion in terms of gambling strategies called martingales. In this formulation, an infinite binary sequences S is random if no lower semicomputable martingale can make unbounded money betting on the successive bits of S
Schnorr and Stimm [31] proved that an infinite binary sequence S is normal if and only if no martingale that is computed by a finite-state automaton can make unbounded money betting on the successive bits of S
Summary
An infinite sequence S over a finite alphabet is normal in the 1909 sense of Borel [7] if every two strings of equal length appear with equal asymptotic frequency in S. The first part of our strong dichotomy theorem says that the infinitely-often exponential rate that can be achieved in 1 above is essentially at least 2Div(S||α)|w|, where w is the prefix of S on which the finite-state gambler has bet so far. The second part of our strong dichotomy theorem says that, if S is α-normal and G is a finite-state gambler betting on S, after each prefix w of S, the capital of G on prefixes w of S is essentially bounded above by 2− RiskG(w). It is to be hoped that our strong dichotomy theorem and the quantitative methods implicit in it will further accelerate these discoveries
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