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Generalized powers and measures

Burdak Zbigniew, Kosiek Marek, Pagacz Patryk, Rudol Krzysztof, Słociński Marek

Opuscula Mathematica

2021

Vol. 41, no. 6

747--754

Using the winding of measures on torus in “rational directions” special classes of unitary operators and pairs of isometries are defined. This provides nontrivial examples of generalized powers. Operators related to winding Szegö-singular measures are shown to have specific properties of their invariant subspaces.

The functional equation and strictly substable random vectors

Borowiecka-Olszewska M.

Probability and Mathematical Statistics

2005

Vol. 25, Fasc. 2

267--278

A random vector X is β-substable, β ∈ (0, 2), if there exist a symmetric β-stable random vector Y and a random variable Θ ≥ 0 independent of Y such that X [symbol] YΘ1/β. In this paper we investigate strictly β-substable random vectors which are generated from a strictly β-stable random vector Y. We study some of their properties. We obtain the theorem that every strictly β-stable random vector X with Θ ∼ Sα/β (σ, 1, 0) is also strictly α-stable, α < β (for the case of random variable X see, e.g., [1], [6]). The opposite theorem is also satisfied, but we obtain something more. We obtain some functional equation and we show that if astrictly β-substable random vector X is α-stable, then it has to be strictly α-stable and the mixing random variable Θ has to have a distribution Sα/β (σ, 1, 0). This is the main result of the paper.

Some remarks on S alpha S, beta - substable random vectors

Misiewicz J.K., Takenaka S.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

253--258

An S α S random vector Xis β-substable, α < β ≤ 2, if Xd = Y Θ 1/β for some symmetric β-stable random vector Y Θ ≥ 0 a random variable with the Laplace transform exp{−tα/β}, Y and Θ are independent. We say that an S α S random vector is maximal if it is not β-substable for any β > α. In the paper we show that the ca,nonical spectral measure for every S α S, β-substablerandom vector X, β > α is equivalent to the Lebesgue measure on Sn−1.We show also that every such vector admits the representation X=Y+Z, where Y is an S α S sub-Gaussian random vector, Z is a maximal S α S random vector, Y and Z are independent. The last representation is not unique.

Individual ergotic theorem for non-contractive normal operators

Jajte R.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

85--99

A condition implying the strong law of large numbers for trajectories of a normal non-contractive operator is given. The condition has been described in terms of a spectral measure, in the spirit of the well-known theorem of V. F. Gaposhkin. To embrace the non-contractive operators we pass from the classical arithmetic (Cesáro) means to the Borel methods of summability.