On a characterization theorem for locally compact abelian groups

The well-known Skitovich-Darmois theorem asserts that a Gaussian distribution is characterized by the independence of two linear forms of independent random variables. The similar result was proved by Heyde, where instead of the independence, the symmetry of the conditional distribution of one linear form given another was considered. In this article we prove that the Heyde theorem on a locally compact Abelian group X remains true if and only if X contains no elements of order two. We describe also all distributions on the two-dimensional torus which are characterized by the symmetry of the conditional distribution of one linear form given another. In so doing we assume that the coefficients of the forms are topological automorphisms of X and the characteristic functions of the considering random variables do not vanish.     

On the Heyde theorem for some locally compact Abelian groups

According to the well-known Heyde theorem Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form in n independent random variables given another. For n = 2 we prove analogs of this theorem in the case when random variables take values in a locally compact Abelian group X, and coefficients of the linear forms are topological automorphisms of the group X.     

On a characterization of a convolution of Gaussian and idempotent distributions

We prove some analogues of the well-known Skitovich-Darmois and Heyde characterization theorems for a second countable locally compact Abelian group X under the assumption that the distributions of the random variables have continuous positive densities with respect to a Haar measure on X and the coefficients in the linear forms considered are topological automorphisms of the group X.     

On a Characterization Theorem for Locally Compact Abelian Groups Containing an Element of Order 2

According to the well-known Heyde theorem the Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. We study analogues of this theorem for some locally compact Abelian groups X containing an element of order 2. We prove that if X contains an element of order 2, this leads to the fact that a wide class of non-Gaussian distributions on X is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given the other. While coefficients of linear forms are topological automorphisms of a group.

     

Central limit theorem for Banach space valued fuzzy random variables

In this paper we prove a central limit theorem for Borel measurable nonseparably valued random elements in the case of Banach space valued fuzzy random variables.

     

Rates of convergence for the laws of large numbers for independent Banach-valued random variables

In this paper, results of Lai, Heyde, and Rohatgi concerning the convergence rates for the laws of large numbers are extended for the case of independent random variables taking values in a separable Banach space.     

Convergence en loi et lois du logarithme itéré pour les vecteurs gaussiens

In this paper we prove a Strassen version of the law of the iterated logarithm for some sequences of weakly asymptotically independant Banach space valued gaussian random variables which converge in distribution, and we prove that the central limit theorem implies the functional form of the law of the iterated logarithm for the partial sums of certain Banach space valued gaussian sequences.Furthermore we give conditions for the convergence in distribution of sequences of gaussian random variables and gaussian stochastic processes, and these conditions permit us to prove that our results generalize in the gaussian case all similar results known to the authors at present.     

Spectral mapping theorem for ascent,essential ascent,descent and essential descent spectrum of linear relations

In [7], Cross showed that the spectrum of a linear relation T on a normed space satisfies the spectral mapping theorem. In this paper, we extend the notion of essential ascent and descent for an operator acting on a vector space to linear relations acting on Banach spaces. We focus to define and study the descent, essential descent, ascent and essential ascent spectrum of a linear relation everywhere defined on a Banach space X. In particular, we show that the corresponding spectrum satisfy the polynomial version of the spectral mapping theorem.     

A fixed point theorem for affine mappings and its application to elasticity theory

In this paper we obtain a general fixed point theorem for an affine mapping in Banach space. As an application of this theorem we study existence of periodic solutions to the equations of the linear elasticity theory.     

A local Hahn–Banach theorem and its applications

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a “local” version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014).     

On the central limit theorem in Banach spaces

It is shown, with the use of a concentration inequality of LeCam, that associated with an infinitely divisible random variable with values in a separable Banach space there is a Lévy-Khintchine formula. A partial converse of this fact is also proved. Relations between the continuity of the compound Poisson and the Gaussian variables associated with a Lévy measure are studied. A central limit theorem is obtained and examples are given.     

On a characterization of convolutions of Gaussian and Haar distributions

We prove some analogues of the well‐known Skitovich–Darmois and Heyde characterization theorems for a second countable locally compact Abelian group X under the assumption that the distributions of the random variables have continuous positive densities with respect to a Haar measure on X and the coefficients in the linear forms considered are topological automorphisms of X.     

THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC DIFFERENTIAL EQUATION IN BANACH SPACES I

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随机集的极限性质

本文研究了Banach空间随机紧集的极限性质.利用嵌入定理,得到了Banach中独立同分布随机紧集的Minkowski和的中偏差原理.     

A non-central banach valued limit theorem and its application

This note is devoted to the generalization of a limit theorem on sums of strongly dependent random variables (non-central limit theorem) to the case when the initial Gaussian sequence, functions of which are considered, assumes values in the Banach space of continuous functions on a compactum; special attention is given to the transition case, when the limit distribution changes by a jump.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 114–119, 1989.     

Multivariate Lukacs theorem

According to the celebrated Lukacs theorem, independence of quotient and sum of two independent positive random variables characterizes the gamma distribution. Rather unexpectedly, it appears that in the multivariate setting, the analogous independence condition does not characterize the multivariate gamma distribution in general, but is far more restrictive: it implies that the respective random vectors have independent or linearly dependent components. Our basic tool is a solution of a related functional equation of a quite general nature. As a side effect the form of the multivariate distribution with univariate Pareto conditionals is derived.     

An operator-valued Lyapunov theorem

We generalize Lyapunov's convexity theorem for classical (scalar-valued) measures to quantum (operator-valued) measures. In particular, we show that the range of a nonatomic quantum probability measure is a weak^?-closed convex set of quantum effects (positive operators bounded above by the identity operator) under a sufficient condition on the non-injectivity of integration. To prove the operator-valued version of Lyapunov's theorem, we must first define the notions of essentially bounded, essential support, and essential range for quantum random variables (Borel measurable functions from a set to the bounded linear operators acting on a Hilbert space).     

An analytic refinement of the Heyde theorem on linear forms of independent random variables

In terms of the characteristic function of the joint distribution of two linear forms of independent random variables, one refines Heyde's known theorem on the characterization of the Gaussian distribution by the property of symmetry of the conditional distribution of one linear form under a given second form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 54–59, 1988.