@MISC{Lutz906adivergence, author = {Jack H. Lutz}, title = {A Divergence Formula for Randomness and Dimension (Short Version)}, year = {906} }

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Abstract

If S is an infinite sequence over a finite alphabet Σ and β is a probability measure on Σ, then the dimension of S with respect to β, written dim β (S), is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension dim(S) when β is the uniform probability measure. This paper shows that dim β (S) and its dual Dim β (S), the strong dimension of S with respect to β, can be used in conjunction with randomness to measure the similarity of two probability measures α and β on Σ. Specifically, we prove that the divergence formula dim β (R) = Dim β (R) =