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

Pratsevityj, M. V.

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.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

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

Keywords:

fractal; Markov chain; spectrum of distribution; singular distribution; Cantor set