scholarly journals Tightness and exponential tightness of Gaussian probabilities

2020 ◽  
Vol 24 ◽  
pp. 113-126
Author(s):  
Paolo Baldi
Keyword(s):  
Banach Space ◽  
Large Deviations ◽  
General Result ◽  
Separable Banach Space ◽  
Simple Criterion ◽  
Exponential Tightness

We prove a simple criterion of exponential tightness for sequences of Gaussian r.v.’s with values in a separable Banach space from which we deduce a general result of Large Deviations which allows easily to obtain LD estimates in various situations.

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

scholarly journals On the convergence of greedy algorithms for initial segments of the Haar basis

2010 ◽  
Vol 148 (3) ◽  
pp. 519-529 ◽  
Author(s):  
S. J. DILWORTH ◽  
E. ODELL ◽  
TH. SCHLUMPRECHT ◽  
ANDRÁS ZSÁK
Keyword(s):  
Banach Space ◽  
Greedy Algorithm ◽  
General Result ◽  
Initial Segment ◽  
Greedy Algorithms ◽  
Dimensional Banach Space ◽  
Haar Basis ◽  
Finite Dimensional ◽  
Finite Dimensional Banach Space ◽  
Number Of Iterations

AbstractWe consider the X-Greedy Algorithm and the Dual Greedy Algorithm in a finite-dimensional Banach space with a strictly monotone basis as the dictionary. We show that when the dictionary is an initial segment of the Haar basis in Lp[0, 1] (1 < p < ∞) then the algorithms terminate after finitely many iterations and that the number of iterations is bounded by a function of the length of the initial segment. We also prove a more general result for a class of strictly monotone bases.

scholarly journals On Common Random Fixed Points of a New Iteration with Errors for Nonself Asymptotically Quasi-Nonexpansive Type Random Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
R. A. Rashwan ◽  
P. K. Jhade ◽  
Dhekra Mohammed Al-Baqeri
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Separable Banach Space ◽  
Iterative Scheme ◽  
Random Fixed Points ◽  
Real Separable Banach Space ◽  
Random Mappings

We prove some strong convergence of a new random iterative scheme with errors to common random fixed points for three and then N nonself asymptotically quasi-nonexpansive-type random mappings in a real separable Banach space. Our results extend and improve the recent results in Kiziltunc, 2011, Thianwan, 2008, Deng et al., 2012, and Zhou and Wang, 2007 as well as many others.

On a Theorem of Beurling and Livingston

1965 ◽  
Vol 17 ◽  
pp. 367-372 ◽  
Author(s):  
Felix E. Browder
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Banach Spaces ◽  
General Result ◽  
Functional Equations ◽  
Linear Functional ◽  
Monotone Mappings ◽  
Conjugate Space ◽  
Non Linear ◽  
Duality Mappings

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

scholarly journals Decompositions of Weakly Compact Valued Integrable Multifunctions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 863 ◽  
Author(s):  
Luisa Di Piazza ◽  
Kazimierz Musiał
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Compact Convex ◽  
Weakly Compact ◽  
Seminal Paper ◽  
Decomposition Property ◽  
Short Overview ◽  
Decomposition Theorems ◽  
State Of Art

We give a short overview on the decomposition property for integrable multifunctions, i.e., when an “integrable in a certain sense” multifunction can be represented as a sum of one of its integrable selections and a multifunction integrable in a narrower sense. The decomposition theorems are important tools of the theory of multivalued integration since they allow us to see an integrable multifunction as a translation of a multifunction with better properties. Consequently, they provide better characterization of integrable multifunctions under consideration. There is a large literature on it starting from the seminal paper of the authors in 2006, where the property was proved for Henstock integrable multifunctions taking compact convex values in a separable Banach space X. In this paper, we summarize the earlier results, we prove further results and present tables which show the state of art in this topic.

Compactness in Lebesgue–Bochner spaces of random variables and the existence of mean-square random attractors

2019 ◽  
Vol 19 (04) ◽  
pp. 1950032
Author(s):  
Yejuan Wang ◽  
Xiangming Zhu ◽  
Peter Kloeden
Keyword(s):  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Separable Banach Space ◽  
Additional Condition ◽  
Probability Space ◽  
Parabolic Partial Differential Equations ◽  
Mean Square ◽  
Probabilistic Argument ◽  
Random Attractors

Let [Formula: see text] be a probability space and let [Formula: see text] be a separable Banach space. It is shown a subset [Formula: see text] of [Formula: see text], where [Formula: see text], is relatively compact in [Formula: see text] if and only if it is uniformly [Formula: see text]-integrable and uniformly tight. The additional condition of scalarly relatively compact required in the literature is shown to hold by a probabilistic argument. The result is then used to establish the existence of a mean-square random attractor for dissipative stochastic differential equations and stochastic parabolic partial differential equations.

scholarly journals Properties of the trajectories of set-valued integrals in banach spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 271-281
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Separable Banach Space ◽  
Hausdorff Metric ◽  
Density Theorem ◽  
Mosco Convergence ◽  
Convexity Theorem ◽  
Convergence Theorems ◽  
Convergence Of Sets ◽  
Differentiability Properties

Let F: T → 2x \ {} be a closed-valued multifunction into a separable Banach space X. We define the sets and We prove various convergence theorems for those two sets using the Hausdorff metric and the Kuratowski-Mosco convergence of sets. Then we prove a density theorem of CF and a corresponding convexity theorem for F(·). Finally we study the “differentiability” properties of K(·). Our work extends and improves earlier ones by Artstein, Bridgland, Hermes and Papageorgiou.

scholarly journals The Topological Support of Gaussian Measure in Banach Space

1975 ◽  
Vol 57 ◽  
pp. 59-63 ◽  
Author(s):  
N. N. Vakhania
Keyword(s):  
Banach Space ◽  
Probability Measure ◽  
Gaussian Measure ◽  
Separable Banach Space ◽  
Covariance Operator ◽  
Probability Measures ◽  
Gaussian Measures ◽  
Linear Spaces

The main result of the present paper is the theorem 1, which describes the topological support of an arbitrary Gaussian measure in a separable Banach space. This theorem will be proved after some discussion of the notion of support itself. But we begin with the reminder of the notion of covariance operator of a probability measure. This notion has a great importance not only for the description of support of Gaussian measures but also for the study of other problems in the theory of probability measures in linear spaces (c.f. [1], [2]).

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