scholarly journals On Decoupling in Banach Spaces

2021 ◽  
Author(s):  
Sonja Cox ◽  
Stefan Geiss
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Random Variables ◽  
Stochastic Integrals ◽  
Moment Inequalities ◽  
Conditional Distributions ◽  
Decoupling Inequalities ◽  
Adapted Process ◽  
Special Case ◽  
Progressive Enlargement

AbstractWe consider decoupling inequalities for random variables taking values in a Banach space X. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar-type expansion in which only the pre-specified conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not affect the decoupling properties (in particular, it does not affect the constants involved). As a special case, we deal with one-sided moment inequalities for decoupled dyadic (i.e., Paley–Walsh) martingales and show that Burkholder–Davis–Gundy-type inequalities for stochastic integrals of X-valued processes can be obtained from decoupling inequalities for X-valued dyadic martingales.

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

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.

RUC-systems and Besselian systems in Banach spaces

1989 ◽  
Vol 106 (1) ◽  
pp. 163-168 ◽  
Author(s):  
D. J. H. Garling ◽  
N. Tomczak-Jaegermann
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Span ◽  
Random Variables ◽  
Biorthogonal System ◽  
Independent Random Variables ◽  
Closed Linear Span ◽  
Image Position

Let (rj) be a Rademacher sequence of random variables – that is, a sequence of independent random variables, with , for each j. A biorthogonal system in a Banach space X is called an RUC-system[l] if for every x in [ej] (the closed linear span of the vectors ej), the seriesconverges for almost every ω. A basis which, together with its coefficient functionals, forms an RUC-system is called an RUC-basis. A biorthogonal system is an RLTC-svstem if and only if there exists 1 ≤ K < ∞ such thatfor each x in [ej]: the RUC-constant of the system is the smallest constant K satisfying (1) (see [1], proposition 1.1).

scholarly journals Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces

2020 ◽  
Vol 379 (2) ◽  
pp. 417-459
Author(s):  
Ivan Yaroslavtsev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Stochastic Integrals ◽  
Reflexive Banach Spaces ◽  
Random Measures ◽  
Umd Spaces ◽  
Independent Increments ◽  
The Right ◽  
Umd Banach Spaces ◽  
Vector Valued

Abstract In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that $$M_0=0$$ M 0 = 0 , we show that the following two-sided inequality holds for all $$1\le p<\infty $$ 1 ≤ p < ∞ : Here $$ \gamma ([\![M]\!]_t) $$ γ ( [ [ M ] ] t ) is the $$L^2$$ L 2 -norm of the unique Gaussian measure on X having $$[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle , \langle M,y^*\rangle ]_t$$ [ [ M ] ] t ( x ∗ , y ∗ ) : = [ ⟨ M , x ∗ ⟩ , ⟨ M , y ∗ ⟩ ] t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of ($$\star $$ ⋆ ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, ($$\star $$ ⋆ ) holds for all $$0<p<\infty $$ 0 < p < ∞ , and that for purely discontinuous martingales the right-hand side of ($$\star $$ ⋆ ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, ($$\star $$ ⋆ ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of ($$\star $$ ⋆ ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

scholarly journals Observations on the Separable Quotient Problem for Banach Spaces

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 7 ◽  
Author(s):  
Sidney A. Morris ◽  
David T. Yost
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Dense Subspace ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Special Cases ◽  
Separable Quotient Problem ◽  
Special Case

The longstanding Banach–Mazur separable quotient problem asks whether every infinite-dimensional Banach space has a quotient (Banach) space that is both infinite-dimensional and separable. Although it remains open in general, an affirmative answer is known in many special cases, including (1) reflexive Banach spaces, (2) weakly compactly generated (WCG) spaces, and (3) Banach spaces which are dual spaces. Obviously (1) is a special case of both (2) and (3), but neither (2) nor (3) is a special case of the other. A more general result proved here includes all three of these cases. More precisely, we call an infinite-dimensional Banach space X dual-like, if there is another Banach space E, a continuous linear operator T from the dual space E * onto a dense subspace of X, such that the closure of the kernel of T (in the relative weak* topology) has infinite codimension in E * . It is shown that every dual-like Banach space has an infinite-dimensional separable quotient.

Some generalizations of the Borsuk–Ulam theorem

1977 ◽  
Vol 82 (1) ◽  
pp. 119-125 ◽  
Author(s):  
D. E. Edmunds ◽  
J. R. L. Webb
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Closed Subspace ◽  
Piecewise Linear ◽  
Closed Unit Ball ◽  
Infinite Dimensional ◽  
Ulam Theorem ◽  
Dimensional Version ◽  
Linear Involution ◽  
Special Case

In a recent paper, Fenn (3) has established the following theorem: if ø is a piecewise linear involution on Sn without fixed points, if f: Sn → Sn and g: are continuous and the degree of f is odd, then there are points x and y on Sn such that f(x) = ø(f(y)) and g(x) = g(y). The Borsuk–Ulam theorem is the special case of this in which f is the identity and ø corresponds to reflexion in the origin. Since an infinite-dimensional version of the Borsuk–Ulam theorem is known, involving compact maps (see, for example, page 72 of (2)), it is natural to ask whether Fenn's result can also be extended to general Banach spaces, and in this paper we give such an extension when ø is reflexion in the origin. More precisely, we prove that if B is the closed unit ball in a Banach space X and f, g: B → X are compact, with deg (I − f, B, 0) odd (this is the Leray–Schauder degree and I is the identity map) and (I − g) (B) contained in a proper, closed subspace of X, then there exist x, y on the boundary ∂B of B and apositive numberα such that α(I − f) (x) = − (I − f) (y) and (I − g) (x) = (I – g) (y).

scholarly journals K-Convexity and Duality for Almost Summing Operators

2000 ◽  
Vol 7 (2) ◽  
pp. 245-268 ◽  
Author(s):  
O. Blasco ◽  
V. Tarieladze ◽  
R. Vidal
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Random Variables ◽  
Convex Banach Space ◽  
Precise Description ◽  
Fixed Sequence ◽  
Almost Summing ◽  
Independent Identically Distributed

Abstract For a fixed sequence f. = (fn ) of independent identically distributed symmetric random variables with , we introduce the notion of Kf. -convex Banach space and the notions of (fn )-bounding and (fn )-converging operators acting between Banach spaces. It is shown that the dual of the space of (fn )-converging operators between a Hilbert space and a Kf. -convex Banach space admits a precise description in terms of trace duality. The obtained results recover similar formulations for almost summing and γ-Radonifying operators.

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

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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