Singularity of the distribution of a random variable represented by an \(A_2\)-continued fraction with independent elements. (Ukrainian, English) Zbl 1224.60021

Prats’ovytyĭ, M. V.; Kyurchev, D. V.

Teor. Jmovirn. Mat. Stat. 81, 139-154 (2009); translation in Theory Probab. Math. Stat. 81, 159-175 (2010).

The authors deal with the random variable \(\xi\) represented by an \(A_2\)-continued fraction \[ \xi=\frac{1}{\eta_1+\frac{1}{\eta_2+\cdots}}\equiv[\eta_1,\eta_2,\dots], \] where \(\eta_k\) are independent random variables with the distribution \(P\{\eta_k=\alpha_1\}=p_{\alpha_1k}\geq0\), \(P\{\eta_k=\alpha_2\}=p_{\alpha_2k}\geq0\), \(0<\alpha_1<\alpha_2\), \(\alpha_1\alpha_2\geq1/2\), \(p_{\alpha_1k}+p_{\alpha_2k}= 1\). An infinite continued fraction \([a_1,a_2,\dots]\), with \(a_k\in A_2=\{\alpha_1,\alpha_2\}\), \(k=1,2,\dots\), \(0 <\alpha_1<\alpha_2\), (called \(A_2\)-continued fraction) is a realization (value) of the random variable \(\xi\) represented by an \(A_2\)-continued fraction. The authors study properties of the distribution (the Lebesgue discrete, absolutely continuous, and singularly continuous components of a distribution) as well as the topological and metric properties of the topological support of the distribution of the random variable \(\xi\) represented by an \(A_2\)-continued fraction. It is proved that the distribution of \(\xi\) cannot be absolutely continuous. A condition is proposed for the distribution of \(\xi\) to belong to one of the two types of singular distributions, Cantor or Salem type, depending on the topological and metric properties of the topological support of the distribution.
For more results and references see [M. Pratsiovytyi and D. Kyurchev, Random Oper. Stoch. Equ. 17, No. 1, 91–101 (2009; Zbl 1224.60020); S. O. Dmytrenko, D. V. Kyurchev and M. V. Prats’ovytyi, Ukr. Mat. Zh. 61, No. 4, 452–463 (2009); translation in Ukr. Math. J. 61, No. 4, 541–555 (2009; Zbl 1224.11066); N. V. Pratsevityj, Ukr. Mat. Zh. 48, No. 8, 1086–1095 (1996; Zbl 0890.60012)].

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60E05	Probability distributions: general theory
11K55	Metric theory of other algorithms and expansions; measure and Hausdorff dimension
11K50	Metric theory of continued fractions
26A30	Singular functions, Cantor functions, functions with other special properties
28A80	Fractals

Keywords:

random continued fraction; \(A_2\)-continued fraction; Cantor type singular distribution; Salem type singular distribution

Citations:

Zbl 0890.60012; Zbl 1224.60020; Zbl 1224.11066