The space $D$ in several variables: random variables and higher moments

We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $\sigma$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions, and we give an application to the Zolotarev distance between two such random functions.

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On Chung-Teicher type strong law for arrays of vector-valued random variables

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204301031 ◽

2004 ◽

Vol 2004 (9) ◽

pp. 443-458

Author(s):

Anna Kuczmaszewska

Keyword(s):

Banach Space ◽

Random Variables ◽

Strong Law ◽

Laws Of Large Numbers ◽

Strong Laws ◽

Large Numbers ◽

Random Elements ◽

Vector Valued

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach spaceℬ. The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of specific series ando(1)requirements on specific weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space.

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Convergence of weighted sums of independent random variables and an extension to Banach space-valued random variables

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171279000272 ◽

1979 ◽

Vol 2 (2) ◽

pp. 309-323

Author(s):

W. J. Padgett ◽

R. L. Taylor

Keyword(s):

Banach Space ◽

Banach Spaces ◽

Random Variables ◽

Almost Sure Convergence ◽

Independent Random Variables ◽

Weighted Sums ◽

Real Numbers ◽

Martingale Theory ◽

Random Elements ◽

Schauder Bases

Let{Xk}be independent random variables withEXk=0for allkand let{ank:n≥1, k≥1}be an array of real numbers. In this paper the almost sure convergence ofSn=∑k=1nankXk,n=1,2,…, to a constant is studied under various conditions on the weights{ank}and on the random variables{Xk}using martingale theory. In addition, the results are extended to weighted sums of random elements in Banach spaces which have Schauder bases. This extension provides a convergence theorem that applies to stochastic processes which may be considered as random elements in function spaces.

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Stability for randomly weighted sums of random elements

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171299225598 ◽

1999 ◽

Vol 22 (3) ◽

pp. 559-568 ◽

Cited By ~ 2

Author(s):

Tien-Chung Hu ◽

Hen-Chao Chang

Keyword(s):

Banach Space ◽

Separable Banach Space ◽

Sufficient Conditions ◽

Random Variables ◽

Necessary And Sufficient Conditions ◽

Weighted Sums ◽

Randomly Weighted Sums ◽

Random Elements ◽

Necessary And Sufficient

Let{Xn:n=1,2,3,…}be a sequence of i.i.d. random elements taking values in a separable Banach space of typepand let{An,i:i=1,2,3,…;n=1,2,3,…}be an array of random variables. In this paper, under various assumptions of{An,i}, the necessary and sufficient conditions for∑i=1∞An,iXi→0a.s. are obtained. Also, the necessity of the assumptions of{An,i}is discussed.

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Higher moments of Banach space valued random variables

Memoirs of the American Mathematical Society ◽

10.1090/memo/1127 ◽

2015 ◽

Vol 238 (1127) ◽

pp. 0-0 ◽

Cited By ~ 3

Author(s):

Svante Janson ◽

Sten Kaijser

Keyword(s):

Banach Space ◽

Random Variables ◽

Higher Moments

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Some results on the span of families of Banach valued independent, random variables

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171291000443 ◽

1991 ◽

Vol 14 (2) ◽

pp. 381-384

Author(s):

Rohan Hemasinha

Keyword(s):

Banach Space ◽

Probability Space ◽

Linear Span ◽

Random Variables ◽

Isomorphic Copy ◽

Independent Random Variables ◽

Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

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Semi-normal operators on uniformly smooth Banach spaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500009356 ◽

1990 ◽

Vol 32 (3) ◽

pp. 273-276 ◽

Cited By ~ 1

Author(s):

Muneo Chō

Keyword(s):

Banach Space ◽

Linear Functional ◽

Dual Space ◽

Complex Banach Space ◽

Linear Operators ◽

Bounded Linear ◽

Uniformly Smooth ◽

Normal Operators ◽

Uniformly Smooth Banach Spaces ◽

The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

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The dual of the space of bounded operators on a Banach space

Concrete Operators ◽

10.1515/conop-2020-0109 ◽

2021 ◽

Vol 8 (1) ◽

pp. 48-59

Author(s):

Fernanda Botelho ◽

Richard J. Fleming

Keyword(s):

Banach Space ◽

Tensor Product ◽

Banach Spaces ◽

Dual Space ◽

Tensor Products ◽

Bounded Operators ◽

Projective Tensor Product ◽

Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

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