Singular and fractal properties of distributions of random variables whose digits of polybasic representations form a homogeneous Markov chain (Q2722128)

Let us take and fix a set of positive numbers \(Q=\{q_{0},\dots, q_{n-1}\},\) where \(n\) is a fixed natural number, \(n\geq 2,\) \(\sum_{i=1}^{n-1}q_{i}=1,\) on the base of which one constructs the set \(A=\{a_{0},\dots, a_{n-1}\}\) such that \(a_{0}=0,\) \(a_{k}=\sum_{i=0}^{k-1}q_{i}.\) For each infinite sequence of numbers \(\{\alpha_{k}\},\) where \(\alpha_{k}\) takes values from the set \(N_{n-1}^{0}=\{0,1,\dots,n-1\},\) let us take the number NEWLINE\[NEWLINEx=a_{\alpha_{1}}+\sum_{k=2}^{+\infty} \Biggl[a_{\alpha_{k}}\prod_{i=1}^ {k-1}q_{\alpha_{i}}\Biggr]= \Delta_{{\alpha_{1}}\dots{\alpha_{k}}\dots}.NEWLINE\]NEWLINE The numbers \(\alpha_{k}\in N_{n-1}^{0}\) are called \(Q\)-digits, and the above expression for \(x\) is called polybasic \(Q\)-representations of the number \(x.\) Fractal properties of distributions of r.v. digits, polybasic \(Q\)-representation (generalization of \(n\)-adic digits) of which form a homogeneous Markov chain for the case where the matrix of transition probabilities contains at least one zero, are studied.