Theory of Probability and Mathematical Statistics
A random variable whose digits in the L-representation have the Markovian dependence
M. V. Prats’ovytyĭ, Yu. V. Khvorostina
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Abstract: The distribution of the random variable θ=1/θ_1+∑_{n=2}^\infty(-1)^{n-1}/(θ_1(θ_1+1)...θ_{n-1}(θ_{n-1}+1)θ_{n-1}) is studied where (θ_n) is a homogeneous Markov chain assuming only positive integer values and having the initial distribution (p_1, p_2, ..., p_n, ...) and transition matrix ||p_{ik}||. The Lebesgue structure of the distribution (discrete, absolutely continuous, and singular components) is studied and topological, metric and fractal properties of the spectrum (the minimal closed support of the distribution) is investigated.
Keywords: Alternating Lüroth series; L-representation; random variable; distribution of the sum of a Lüroth series whose terms are random variables with the Markov dependence; Lebesgue structure of distributions; singular distribution with an anomalous fractal spectrum
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