Plichko A. M.
Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 506-518
We study the conditions for the weak convergence of the maximum of sums of independent random processes in the spaces $C[0, 1]$ and $L_p$ and present examples of applications to the analysis of statistics of the type $\omega 2 $.
Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 504–513
We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.
Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 653–665
For a sequence of independent random elements in a Banach space, we obtain an upper bound for moments of the supremum of normalized sums in the law of the iterated logarithm by using an estimate for moments in the law of large numbers. An example of their application to the law of the iterated logarithm in Banach lattices is given.
Limit Theorems for Random Elements in Ideals of Order-Bounded Elements of Functional Banach Lattices
Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 41-49
For a sequence of independent random elements belonging to an ideal of order-bounded elements of a Banach lattice, we investigate the asymptotic relative stability of extremal values, the law of large numbers for the pth powers, and the central limit theorem.
Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 46–55
In the same way as the known spacesM p ,M p , andI p are constructed on the basis of the spaceL p (−1, 1), we construct the corresponding “limit” spacesM E ,M E , andI E on the real line on the basis of a symmetric function spaceE on a segment and study some of their Banach properties.
On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space
Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1225–1231
Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ 1 n biXi, and Bn=∑ 1 n b i 2 . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (Sn/B n ln ln Bn)1/2).
Ukr. Mat. Zh. - 1992. - 44, № 8. - pp. 1129–1132
The automatic continuity of a linear multiplicative operator T: X?Y, where X and Y are real complete metrizable algebras and Y semi-simple, is proved. It is shown that a complex Frechét algebra with absolute orthogonal basis (xi) (orthogonal in the sense that xiXj=0 if i ? j) is a commutative symmetric involution algebra. Hence, we are able to derive the well-known result that every multiplicative linear functional defined on such an algebra is continuous. The concept of an orthogonal Markushevich basis in a topological algebra is introduced and is applied to show that, given an arbitrary closed subspace Y of a separable Banach space X, a commutative multiplicative operation whose radical is Y may be introduced on X. A theorem demonstrating the automatic continuity of positive functionals is proved.
Ukr. Mat. Zh. - 1990. - 42, № 6. - pp. 777–781