# Plichko A. M.

### Limit theorems for the maximum of sums of independent random processes

Matsak I. K., Plichko A. M., Sheludenko A. S.

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 $.

### On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

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.

### One moment estimate for the supremum of normalized sums in the law of the iterated logarithm

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 *p*th powers, and the central limit theorem.

### Limits on the real line of symmetric spaces on segments

Ukr. Mat. Zh. - 1995. - 47, № 1. - pp. 46–55

In the same way as the known spaces*M* _{ p },M _{ p }, and*I* _{ p } are constructed on the basis of the space*L* _{ p }(−1, 1), we construct the corresponding “limit” spaces*M* _{ E },M _{ E }, and*I* _{ E }on the real line on the basis of a symmetric function space*E* 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, S_{n}= ∑ _{1} ^{n} b_{i}X_{i}, and B_{n}=∑ _{1} ^{n} b _{i} ^{2} . Under certain moment restrictions imposed on the variables*X* _{n}, the conditions for the growth of the sequence (b_{n}) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (S_{n}/*B* _{n} ln ln B_{n})^{1/2}).

### Automatic continuity, bases, and radicals in metrizable algegbras

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 (x_{i}) (orthogonal in the sense that x_{i}X_{j}=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.

### Theory of regularizability in topological vector spaces

Ukr. Mat. Zh. - 1990. - 42, № 6. - pp. 777–781