Незалежність цифр $Q_2$-зображення випадкової величини з заданим розподілом

Abstract

Let $\xi$ be a random variable with a given (uniform, exponential.) probability distribution on a segment $[0;1]$. We study
conditions for $Q_2$-digits $(\xi_n)$ of random variable
$\xi=\Delta^{Q_2}_{\xi_1\xi_2...\xi_n...}$ to be independent. For $\xi$
with exponential distribution, we prove that digits are independent if
and only if parameters $q_0$ and $q_1$ of this system of
representation are equal to $\frac12$. Otherwise digits are dependent
and this dependence is more complicated than Markov dependence. If the function of distribution of random variable with independent $Q_2$-digits has a positive derivative at all $Q_2$-binary points, then its distribution is uniform or exponential, moreover in the latter case the $Q_2$-representative is binary.