Properties of distributions of random variables with independent differences of sequential elements of Ostrogradsky series. (Ukrainian, English) Zbl 1074.60014

Prats’ovytyj, M. V.; Baranovs’kyj, O. M.

Teor. Jmovirn. Mat. Stat. 70, 131-143 (2004); translation in Theory Probab. Math. Stat. 70, 147-160 (2005).

The Ostrogradsky series of the first kind is the series \[ q_0+\frac{1}{q_1}-\frac{1}{q_1q_2}+\cdots+\frac{(-1)^{n-1})}{q_1q_2\cdots q_n}+\cdots, \] where \(q_0\) is an integer number and \(q_1,q_2,\dots\) are natural numbers, \(q_{k+1}>q_k,k\in\mathbb N\). This series is denoted as \(O_1(q_0;q_1,q_2,\dots,q_n,\dots)\). Every real number \(x\in \mathbb R\) can be represented in the form of the Ostrogradsky series which is called \(O_1\)-representation. Representation of a real number \(x\in \mathbb R\) in the form \[ x=q_0+\frac{1}{g_1}-\frac{1}{g_1(g_1+g_2)}+\cdots+\frac{(-1)^{n-1})}{g_1(g_1+g_2)\cdots (g_1+g_2+\cdots+g_n)}+\cdots, \] where \(g_1=q_1,g_{n+1}=q_{n+1}-q_n,\) is called \(\overline{O}_1\)-representation and denoted as \(\overline{O}_1(q_0;g_1,g_2,\dots,\) \(g_n,\dots)\), \(g_k\) is called \(\overline{O}_1\)-element of the number \(x\). The authors deal with the random variables \(\xi=\overline{O}_1(0;\eta_1,\eta_2,\dots,\eta_n,\dots)\), where \(\overline{O}_1\)-elements \(\eta_k\) are independent random variables that take values \(1,2,\dots,m\dots\) with probabilities \(p_{1k},p_{2k},\dots,p_{mk},\dots\), respectively. It is proved that the distribution of such random variables \(\xi\) is pure (pure discrete or pure singular, or pure continuous).

MSC:

60E05	Probability distributions: general theory
26A30	Singular functions, Cantor functions, functions with other special properties
11A67	Other number representations
11K55	Metric theory of other algorithms and expansions; measure and Hausdorff dimension

Keywords:

discrete distribution; singular distribution; continuous distribution