## Fractal properties of distributions of Cantor-type random variables whose $$Q$$-digits form a homogeneous Markov chain.(English. Ukrainian original)Zbl 0941.60083

Theory Probab. Math. Stat. 58, 151-160 (1999); translation from Teor. Jmovirn. Mat. Stat. 58, 139-148 (1998).
Let $$x=\Delta_{\alpha_1\ldots\alpha_{k}\ldots}$$ be the $$Q$$-representation of a number $$x\in[0,1],$$ that is $x=\Delta_{\alpha_1\ldots\alpha_{k}\ldots}= a_{\alpha_1(x)}+\sum_{k=2}^{\infty} \Biggl[a_{\alpha_{k}(x)}\prod_{j=1}^{k-1} q_{\alpha_{j}(x)}\Biggr],$ where $$0<q_{i}\in Q=\{ q_0,\ldots,q_{n-1}\}$$, $$n\geq 2$$, is a fixed natural number, $$\sum_{i=0}^{n-1} q_{i}=1$$, $$a_0=0$$, $$a_{k}=\sum_{i=0}^{k-1} q_{i}$$, $$N_{n-1}^0=\{ 0,1,\ldots,n-1\} \ni \alpha_{k}=\alpha_{k}(x)$$ is $$k$$th $$Q$$-mark of $$x.$$ The author considers a random variable $$\xi=\xi(\|p_{ik}\|) =\Delta_{\eta_1\ldots\eta_{k}\ldots}$$, such that $$Q$$-marks $$\eta_{k}$$ form a Markov chain with initial probabilities $$p_0,p_1,\ldots,p_{n-1}$$ ($$p_{i}>0$$) and with a matrix of transitive probabilities $$\|p_{ik}\|$$ $$(p_{ik}\geq 0)$$, $$i,k\in N_{n-1}^0.$$ The author investigates the structure of the distribution of $$\xi$$ and the fractal properties of the spectrum of the distribution of $$\xi.$$ For example, the following assertion is proved. The distribution of a random variable $$\xi$$ is a singular distribution of Cantor type if and only if it is continuous and the matrix of transitive probabilities contains at least one zero.

### MSC:

 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60E05 Probability distributions: general theory