Theory of Probability and Mathematical Statistics
Properties and distributions of values of fractal functions related to Q2-representation of real numbers
M. V. Pratsiovytyi, S. P. Ratushniak
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Abstract: We consider Q2-representation of numbers x∈[0,1] defined by one parameter q0∈(0;1) and expansion of numbers x∈[0,1] in seriesx=α1q1-α1+∑∞k=2(αkq1-αk∏k-1j=1qαj(x))≡ΔQ2α1α2...αn... ,
where α
k∈{0,1}≡A, q
1≡1−q
0. We study structural, local and global topological, metric and fractal properties of the function defined by equality
fφ(x)=fφ(ΔQ2α1α2α3...αn-1αnαn+1...)=ΔQ2φ(α1α2)(α2α3)...φ(αn-1αn)(αnαn+1)... ,
where φ is a given function (φ: A
2 → A).
Let X be a random variable with a given distribution. For random variable Y=F(X), Lebesgue structure (i.e., content of discrete, absolutely continuous and singular components) and spectral properties (properties of the set of points of increasing for the probability distribution function) are studied.
Keywords: Q2-representation of fractional part of real number, classic binary representation of number, shift operator of digits of representation, inversor of digits of representation, singular function, fractal functions, level set of function, distribution of values of function.
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