Abstract
For the simplest colored branching process, we prove an analog to the McMillan theorem and calculate the Hausdorff dimensions of random fractals defined in terms of the limit behavior of empirical measures generated by finite genetic lines. In this setting, the role of Shannon’s entropy is played by the Kullback–Leibler divergence, and the Hausdorff dimensions are computed by means of the so-called Billingsley–Kullback entropy, defined in the paper.
Highlights
Let us fix a finite set Ω = {1, . . . , r}, whose elements denote different colors, and consider a finite random set X containing a random number of elements of colors 1, . . . , r
We investigate Hausdorff dimensions of random fractals defined in terms of the limit behavior of the sequence of empirical measures δx,n
Let us define a sequence of random sets Xn ⊂ Ωn in the following way: each Xn contains each of 2n strings x = (x1, . . . , xn ) ∈ Ωn with probability p(x1 ) · · · p(xn )
Summary
The value of ρ(ν, μ) coincides with the usual Kullback–Leibler divergence and differs from it only in the fact that in our setting, the measure μ is not a probability, and so ρ(ν, μ) can be negative This is the first main result of the paper. We investigate Hausdorff dimensions of random fractals defined in terms of the limit behavior of the sequence of empirical measures δx,n. Ω , where Ω is a finite set, defined in terms of the limit behavior of empirical measures He investigated Hausdorff dimensions of such fractals and, in particular, obtained formulas analogous to Formulars (2) and (4) (with ΩN instead of X∞ and μ(i) ≡ 1). The present research combines both features of random self-similar fractals and subsets of nonrandom self-similar fractals defined in terms of the limit behavior of empirical measures The second part (Sections 2 and 8–11) contains essentially new results, which have been briefly formulated above
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