Necessary and sufficient conditions for complete convergence of double weighted sums of pairwise independent identically distributed random elements in Banach spaces

2018 ◽  
Vol 157 (2) ◽  
pp. 312-326
Author(s):  
L. V. Thanh ◽  
N. T. Thuy
Keyword(s):  
Banach Spaces ◽  
Complete Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weighted Sums ◽  
Random Elements ◽  
Necessary And Sufficient ◽  
Independent Identically Distributed

scholarly journals Stability for randomly weighted sums of random elements

1999 ◽  
Vol 22 (3) ◽  
pp. 559-568 ◽  
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.

MAZUR–ULAM PROPERTY OF THE SUM OF TWO STRICTLY CONVEX BANACH SPACES

2015 ◽  
Vol 93 (3) ◽  
pp. 473-485 ◽  
Author(s):  
JIAN-ZE LI
Keyword(s):  
Banach Spaces ◽  
Equivalent Form ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Extension Problem ◽  
Isometric Extension ◽  
Strictly Convex ◽  
Necessary And Sufficient ◽  
Strictly Convex Banach Spaces ◽  
Equivalent Conditions

In this article, we study the Mazur–Ulam property of the sum of two strictly convex Banach spaces. We give an equivalent form of the isometric extension problem and two equivalent conditions to decide whether all strictly convex Banach spaces admit the Mazur–Ulam property. We also find necessary and sufficient conditions under which the $\ell ^{1}$-sum and the $\ell ^{\infty }$-sum of two strictly convex Banach spaces admit the Mazur–Ulam property.

scholarly journals Balls intersection properties of Banach spaces

1992 ◽  
Vol 45 (2) ◽  
pp. 333-342 ◽  
Author(s):  
Dongjian Chen ◽  
Zhibao Hu ◽  
Bor-Luh Lin
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Necessary And Sufficient Conditions ◽  
Asplund Space ◽  
Necessary And Sufficient

Necessary and sufficient conditions for a Banach space with the Mazur intersection property to be an Asplund space are given. It is proved that Mazur intersection property is determined by the separable subspaces of the space. Corresponding problems for a space to have the ball-generated property are considered. Some comments on possible renorming so that a space having the Mazur intersection property are given.

scholarly journals Datko criteria for uniform instability in Banach spaces

2021 ◽  
Vol 66 (1) ◽  
pp. 115-122
Author(s):  
Rovana Boruga Toma ◽  
Mihail Megan
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Evolution Operators ◽  
Necessary And Sufficient

The main objective of this paper is to present some necessary and sufficient conditions of Datko type for the uniform exponential and uniform polynomial instability concepts for evolution operators in Banach spaces.

scholarly journals A unifying approach for some nonexpansiveness conditions on modular vector spaces

2020 ◽  
Vol 25 (5) ◽  
Author(s):  
Andrea Bejenaru ◽  
Mihai Postolache
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Vector Spaces ◽  
Minimizing Sequences ◽  
Ishikawa Iteration ◽  
New Class ◽  
Condition C ◽  
Necessary And Sufficient

This paper provides a new, symmetric, nonexpansiveness condition to extend the classical Suzuki mappings. The newly introduced property is proved to be equivalent to condition (E) on Banach spaces, while it leads to an entirely new class of mappings when going to modular vector spaces; anyhow, it still provides an extension for the modular version of condition (C). In connection with the newly defined nonexpansiveness, some necessary and sufficient conditions for the existence of fixed points are stated and proved. They are based on Mann and Ishikawa iteration procedures, convenient uniform convexities and properly selected minimizing sequences.

On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

1999 ◽  
Vol 36 (1) ◽  
pp. 78-85 ◽  
Author(s):  
M. S. Sgibnev
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Stationary Distributions ◽  
Necessary And Sufficient ◽  
Submultiplicative Function ◽  
Independent Identically Distributed

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.

Right and Left Weak Approximation Properties in Banach Spaces

2009 ◽  
Vol 52 (1) ◽  
pp. 28-38 ◽  
Author(s):  
Changsun Choi ◽  
Ju Myung Kim ◽  
Keun Young Lee
Keyword(s):  
Banach Spaces ◽  
Approximation Property ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weak Approximation ◽  
Weak Version ◽  
Approximation Properties ◽  
Necessary And Sufficient

AbstractNew necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

scholarly journals FRÉCHET FRAMES, GENERAL DEFINITION AND EXPANSIONS

2014 ◽  
Vol 12 (02) ◽  
pp. 195-208 ◽  
Author(s):  
STEVAN PILIPOVIĆ ◽  
DIANA T. STOEVA
Keyword(s):  
Banach Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sequence Spaces ◽  
General Definition ◽  
Fréchet Space ◽  
Frechet Space ◽  
Bk Space ◽  
Frame Expansions ◽  
Necessary And Sufficient

We define an (X1, Θ, X2)-frame with Banach spaces X2 ⊆ X1, ‖ ⋅ ‖1 ≤ ‖ ⋅ ‖2, and a BK-space [Formula: see text]. Then by the use of decreasing sequences of Banach spaces [Formula: see text] and of sequence spaces [Formula: see text], we define a General Fréchet frame on the Fréchet space [Formula: see text]. We obtain frame expansions of elements of XF and its dual [Formula: see text], as well of some of the generating spaces of XF with convergence in appropriate norms. Moreover, we determine necessary and sufficient conditions for a General pre-Fréchet frame to be a General Fréchet frame, as well as for the complementedness of the range of the analysis operator U : XF → ΘF. Several examples illustrate our investigations.

scholarly journals Complete Convergence for Maximal Sums of Negatively Associated Random Variables

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Victor M. Kruglov
Keyword(s):  
Complete Convergence ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Associated Random Variables ◽  
Negatively Associated ◽  
Maximal Inequalities ◽  
Negatively Associated Random Variables ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for the complete convergence of maximal sums of identically distributed negatively associated random variables. The conditions are expressed in terms of integrability of random variables. Proofs are based on new maximal inequalities for sums of bounded negatively associated random variables.

scholarly journals Approximation on the sphere by weighted Fourier expansions

2005 ◽  
Vol 2005 (4) ◽  
pp. 321-339 ◽  
Author(s):  
V. A. Menegatto ◽  
A. C. Piantella
Keyword(s):  
Spherical Harmonics ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Important Case ◽  
Weighted Sums ◽  
Main Theme ◽  
Fourier Expansions ◽  
Necessary And Sufficient

The main theme of this paper is the approximation on the sphere by weighted sums of spherical harmonics. We give necessary and sufficient conditions on the weights for convergence in both the continuous and theLPcases. Approximation by spherical convolution is a particular and important case that fits into our setting.

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