Distribution of unit mass on one fractal self-similar web-type curve

Abstract

In the article, we study structural, spectral, topological, metric and fractal properties of distribution of complex-valued random variable (Formula presented), where (τn) is a sequence of independent random variables taking the values 0, 1, · · ·, 6 with the probabilities p0n, p1n, · · ·, p6n; ε6 = 0, ε0, ε1, · · ·, ε5 are 6th roots of unity. We prove that the set of values of random variable τ is self-similar six petal snowflake which is a fractal curve G of spider web type with dimension log3 7. Its outline is the Koch snowflake. We establish that τ has either a discrete or a singularly continuous distribution with respect to two-dimensional Lebesgue measure. The criterion of discreteness for the distribution is found and its point spectrum (set of atoms) is described. It is proved that the point spectrum is a countable everywhere dense set of values of the random variable τ, which is the tail set of the seven-symbol representation of the points of the curve G. In the case of identical distribution of the random variables τn (namely: pkn = pk) we establish that the spectrum of distribution τ is a self-similar fractal and that the essential support of density is the fractal Besicovitch-Eggleston type set. The set is defined by terms digits frequencies and has the fractal dimension (Formula presented) with respect to the Hausdorff-Billingsley α-measure. The measure is a probabilistic generalization of the Hausdorff α-measure. In this case, the random variables (Formula presented) and (Formula presented) defined by different probability vectors (p0, · · ·, p6) and (p′0, · · ·, p′6) have mutually orthogonal distributions.