Asymptotic Divergences and Strong Dichotomy

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. asserts that, for any such sequence S, the following two things are true. (1) If S is not normal in sense Borel (meaning every strings equal length appear with asymptotic frequency S), then there gambler wins money at an infinitely-often exponential rate betting S. (2) normal, loses In this paper we use Kullback-Leibler divergence to formulate lower div(S||α) probability measure α Σ over upper Div(S||α) way α-normal string w has α(w) S) if only Div(S||α)=0. We also quantify total risk Risk G (w) G takes when along prefix Our main strong uses above notions rates winning losing sides (with latter routinely extended normality α-normality). Modulo caveats paper, our says hold prefixes ( $1~'$ ) 1 2 Div(S||α)|w| . $2~'$ loss xmlns:xlink="http://www.w3.org/1999/xlink">- RiskG(w) (1 $'$ show 1- Div(S||α)/c, where c = log(1/ min xmlns:xlink="http://www.w3.org/1999/xlink">a ∈ Σ α(a)), bound α-dimension prove dual fact div(S||α)/c