Nikoloz Vakhania

A criterion is proved for unconditional convergence of a specific series constructed by Hadamard matrices in the Banach spaces l_p.

For any real normed space X and each positive m there exists a weakly subgaussian random vector with values in X such that $E ln^(m)(||\xi||)=\infty$, ln^(m) being the m-th iteration of natural logarithm 

In Banach spaces l_p, for p\geq 2 a series of a specific type, Sylvester series, converges only if it converges unconditionally. For 1\leq p < 2 there eixist Sylvester series that converge but fail to converge unconditionally

A survey of properties of weakly sub-Gaussian random elements in infinite-dimensional spaces is given.Some new results and examples are also presented

We clarify the connection between (1) regular conditional probabilities relative to sigma-algebra, (2) regular conditional probabilities relative to a mapping and (3) disitntegration with respect to mapping

Characterizations of the class of Banach spaces isomorphing to the space $c_0$, as well as to the class of Banach spaces not containing $l_\infty^n$'s uniformly, are obtained in terms of almost surely unconditional convergence of weakly sub-Gaussian random series. Under almost surely unconditional convergence of random series, convergence of all permutations on the same set of full probability is understood. The questions of almost surely unconditional and weak absolute convergence in the spaces isomorphing to $c_0$ are investigated as well

It is shown that a Gaussian measure in a given infinite-dimensional Banach space always admits an essentially unique Gaussian disintegration with respect to a given continuous linear operator. This covers a similar statement made earlier in [Lee and Wasilkowski, Approximation of linear functionals on a Banach space with a Gaussian measure, J. Complexity 2(1) (1986) 12–43.] for the case of finite-rank operators

The two basic theorems of characterization of Gaussian (normal) distributions are those of G. Pólya's [Math. Zs. 18, 96–108 (1923; JFM 49.0366.01)] and Skitovich-Darmais'. The first of this two can be proved by the elementary methods based on the Lévy continuity theorem, while the second one uses non-elementary and non-statistical tools from the theory of analytic functions. We give an elementary proof of Skitovich-Darmais' theorem for the case of two random variables in each of the two linear forms. The proof is based on the reduction of Skitovich-Darmais' theorem to Pólya's theorem.

Characterization of the Banach spaces isomorphic to the Banach space 0 c is obtained in terms of unconditionally converging series

We give the solution of the general version of Lewis Carroll’s probability problem No. 72 for any number of black or white counters initially in the bag, and any number of non-random black or white counters that could be put into the bag in addition. The result for the general version is given as formula (4).

The main result of this paper is the formulation and proof of Polya's characterization theorem of Gaussian random variables with values in quaternion algebras. We begin with some preliminary information about quaternions and quaternion random variables. More complete information about quaternions (except the items connected with the jointly quaternion systems) can be found in [N. N. Vakhania, Theory Probab. Appl., 43 (1999), pp. 99–115]. Closely related to the present paper is also the paper [N. N. Vakhania, J. Complexity, 13 (1997), pp. 480–488] in which Polya's characterization theorem for complex random variables is proved.

In the present paper we construct an example of a quaternion random variable such that Polya's type characterization theorem of Gaussian distributions does not hold. The matter is that in the linear form, consisting of the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and the other part on the left-hand side. This gives a negative answer to the question posed in [Vakhania and Chelidze, Teor. Veroyatnost. i Primenen. 54: 337–344, 2009]

In the present note we show that Polya’s type characterization theorem of Gaussian distributions does not hold. This happens because in the linear form, constituted by the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and another part on the left side. This gives a negative answer to the question posed in [1]

We give an elementary proof of Skitovich-Darmois theorem for the cases of two random variables in each of the two linear forms, when the random variables take values in complex or quaternion fields. The proof is based on the reduction of Skitovich-Darmois theorem to Polya's theorem in complex and quaternion cases respectively.

In this paper a sufficient and a necessary condition for unconditional convergence of a series in a Banach space with an unconditional basis are analyzed.