On relatively uniform convergence of weighted sums of -lattice valued random elements
On shifted convolution powers of a probility measure.
On Skorohod embedding in -dimensional brownian motion by means of natural stopping times
On small deviation probabilities of positive random variables.
On some conjecture concerning Gaussian measures of dilatations of convex symmetric sets
The paper deals with the following conjecture: if μ is a centered Gaussian measure on a Banach space F,λ > 1, K ⊂ F is a convex, symmetric, closed set, P ⊂ F is a symmetric strip, i.e. P = {x ∈ F : |x'(x)| ≤ 1} for some x' ∈ F', such that μ(K) = μ(P) then μ(λK) ≥ μ(λP). We prove that the conjecture is true under the additional assumption that K is "sufficiently symmetric" with respect to μ, in particular it is true when K is a ball in Hilbert space. As an application we give estimates of Gaussian...
On some contributions to quantum structures by fuzzy sets
It is well known that the fuzzy sets theory can be successfully used in quantum models ([5, 26]). In this paper we give first a review of recent development in the probability theory on tribes and their generalizations – multivalued (MV)-algebras. Secondly we show some applications of the described method to develop probability theory on IF-events.
On some definition of expectation of random element in metric space
We are dealing with definition of expectation of random elements taking values in metric space given by I. Molchanov and P. Teran in 2006. The approach presented by the authors is quite general and has some interesting properties. We present two kinds of new results:• conditions under which the metric space is isometric with some real Banach space;• conditions which ensure "random identification" property for random elements and almost sure convergence of asymptotic martingales.
On Strassen's theorem on stochastic domination.
On strong laws of large numbers for arrays of rowwise independent random elements.
On strong liftings for projective limits
We discuss the permanence of strong liftings under the formation of projective limits. The results are based on an appropriate consistency condition of the liftings with the projective system called "self-consistency", which is fulfilled in many situations. In addition, we study the relationship of self-consistency and completion regularity as well as projective limits of lifting topologies.
On subsets of and -stable processes
On tails of stationary measures on a class of solvable groups
On the almost sure convergence of weighted sums of random elements in D(0,1).
On the Argmin-sets of stochastic processes and their distributional convergence in Fell-type-topologies
Let be the collection of all -optimal solutions for a stochastic process with locally bounded trajectories defined on a topological space. For sequences of such stochastic processes and of nonnegative random variables we give sufficient conditions for the (closed) random sets to converge in distribution with respect to the Fell-topology and to the coarser Missing-topology.
On the best constant in the Khinchin-Kahane inequality
We prove that if is the Rademacher system of functions then for any sequence of vectors in any normed linear space F.
On the bounded laws of iterated logarithm in Banach space
In the present paper, by using the inequality due to Talagrand's isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given.
On the bounded laws of iterated logarithm in Banach space
In the present paper, by using the inequality due to Talagrand’s isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given.
On the bundle convergence of double orthogonal series in noncommutative -spaces
The notion of bundle convergence in von Neumann algebras and their -spaces for single (ordinary) sequences was introduced by Hensz, Jajte, and Paszkiewicz in 1996. Bundle convergence is stronger than almost sure convergence in von Neumann algebras. Our main result is the extension of the two-parameter Rademacher-Men’shov theorem from the classical commutative case to the noncommutative case. To our best knowledge, this is the first attempt to adopt the notion of bundle convergence to multiple series....
On the central limit theorem in Hilbert space.
The object of this paper is to prove a central limit theorem in (separable) Hilbert space using a method based on the so called découpage de Lévy, the Lindeberg proof for the Gaussian case and an elementary proof of Poisson convergence for the direct part, and on elementary probabilistic inequalities for the converse. In particular, characteristic functions are only used in unicity questions. Several results of Varadhan (1962) can be obtained either directly as corollaries of the main theorem or...
On the characterization of isotropic Gaussian fields on homogeneous spaces of compact groups.