Meta-intransitive Systems of Independent Random Variables and Fractals

We construct meta-intransitive systems of independent random variables of any finite order from basic tuple of random variables which generalize intransitive dice. Under this construction, the equality of some linear functional is preserved, provided this equality hold for the basic variables. This scheme is also extended to infinite-order systems, which are shown to be fractals, and for which the similarity dimension can be evaluated. The problem of the upper bound for this dimension is posed.