Asymptotic divergences and strong dichotomy

The Schnorr-Stimm dichotomy theorem [31] concerns finite-state gamblers that bet on infinite sequences of symbols taken from a finite alphabet S. 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 betting on S 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||a) of a probability measure a on S from a sequence S over S and the upper asymptotic divergence Div(S||a) of a from S in such a way that a sequence S is a-normal (meaning that every string w has asymptotic frequency a(w) in S) if and only if Div(S||a) = 0. We also use the Kullback-Leibler divergence to quantify the total risk RiskG(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 a-normality). Modulo asymptotic caveats in the paper, our strong dichotomy theorem says that the following two things hold for prefixes w of S. (10) The infinitely-often exponential rate of winning in 1 is 2Div(S||a)|w| . (20) The exponential rate of loss in 2 is 2-RiskG(w) . We also use (10) to show that 1 - Div(S||a)/c, where c = log(1/mina¿S a(a)), is an upper bound on the finite-state a-dimension of S and prove the dual fact that 1 - div(S||a)/c is an upper bound on the finite-state strong a-dimension of S

Dimensions of Copeland-Erdös Sequences

The base-k Copeland-Erdös sequence given by an infinite set A of positive integers is the infinite sequence CEk(A) formed by concatenating the base-k representations of the elements of A in numerical order. This paper concerns the following four quantities. • The finite-state dimension dimFS(CEk(A)), a finite-state version of classical Hausdorff dimension introduced in 2001. • The finite-state strong dimension DimFS(CEk(A)), a finite-state version of classical packing dimension introduced in 2004. This is a dual of dimFS(CEk(A)) satisfying DimFS(CEk(A)) ≥ dimFS(CEk(A)). • The zeta-dimension Dimζ(A), a kind of discrete fractal dimension discovered many times over the past few decades. • The lower zeta-dimension dimζ(A), a dual of Dimζ(A) satisfying dimζ(A) ≤ Dimζ(A). We prove the following. 1. dimFS(CEk(A)) ≥ dimζ(A). This extends the 1946 proof by Copeland and Erdös that the sequence CEk(PRIMES) is Borel normal. 2. DimFS(CEk(A)) ≥ Dimζ(A). 3. These bounds are tight in the strong sense that these four quantities can have (simultaneously) any four values in [0, 1] satisfying the four above-mentioned inequalities