Elements In Banach Spaces Research Articles

On conditional compactly uniform pth-order integrability of random elements in Banach spaces

The notion of conditional compactly uniform pth-order integrability of an array of random elements in a separable Banach space concerning an array of random variables and relative to a sequence of σ-algebras is introduced and characterized. We state a conditional law for randomly weighted sums of random elements in a Banach space with the bounded approximation property, and we prove that, under the introduced condition, the problem can be reduced to a similar problem for random elements in a finite-dimensional space.

An application of the Ryll-Nardzewski–Woyczyński theorem to a uniform weak law for tail series of weighted sums of random elements in Banach spaces

For a sequence of Banach space valued random elements { V n , n⩾1} (which are not necessarily independent) with the series ∑ n=1 ∞ V n converging unconditionally in probability and an infinite array a={a ni, i⩾n, n⩾1} of constants, conditions are given under which (i) for all n⩾1, the sequence of weighted sums ∑ i=n m a niV i converges in probability to a random element T n ( a) as m→∞, and (ii) T n(a) → P 0 uniformly in a as n→∞ where a is in a suitably restricted class of infinite arrays. The key tool used in the proof is a theorem of Ryll-Nardzewski and Woyczyński (1975, Proc. Amer. Math. Soc. 53, 96–98).

On convergence of series of independent random elements in banach spaces

The rate of convergence for an almost certainly convergent series independent random elements in a real separable Banach space is studied in this paper. More specifically, when S n converges almost certainly to a random element S, the tail series is a well defined sequence of random elements with almost certainly. The main result establishes for a sequence of positive constants with b j ⩽ Const. b n whenever the equivalence between the tail series weak law of large numbers and the limit law thereby extending a result of Nam and Rosalsky [20] to a Banach space setting while also simplifying the argument used in the earlier result. The quasimonotonicity proviso on cannot be dispensed with

On complete convergence for arrays of rowwise independent random elements in banach spaces

We extend and generalize some recent results on complete convergence (cf. Hu, Moricz, and Taylor [14], Gut [11], Wang, Bhaskara Rao, and Yang [26], Kuczmaszewska and Szynal [17], and Sung [23]) for arrays of rowwise independent Banach space valued random elements. In the main result, no assumptions are made concerning the existence of expected values or absolute moments of the random elements and no assumptions are made concerning the geometry of the underlying Banach space. Some well-known results from the literature are obtained easily as corollaries. The corresponding convergence rates are also established

Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces

Strong laws of large numbers for arrays of orthogonal random elements in Banach spaces

Marcinkiewicz strong laws of large numbers and convergence rates for arrays of independent random elements in banach spaces

Consider a set of independent Banach space-valued random elements indexed by the positive integer d-dimensional lattice points, d ≥ 2. The classical Kolmogorov-Marcinkiewicz strong law of large numbers is obtained in this setting. In addition, a rate of convergence for the tall probabilities of the sample sums is determined

Estimates of the Distance between Sums of Independent Random Elements in Banach Spaces

Previous article Next article Estimates of the Distance between Sums of Independent Random Elements in Banach SpacesV. Yu. Bentkus and A. RachkauskasV. Yu. Bentkus and A. Rachkauskashttps://doi.org/10.1137/1129005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] V. M. Zolotarev, Metric distances in spaces of random variables and of their distributions, Math. USSR Sb., 30 (1976), 373–401 0383.60022 CrossrefGoogle Scholar[2] V. M. Zolotarev, Approximation of the distributions of sums of independent random variables with values in infinite-dimensional spaces, Theory Prob. Appl., 21 (1976), 721–737, correction in 22 (1977), p. 881 0378.60003 LinkGoogle Scholar[3] A. S. Nemirovskii and , S. M. Semenov, On polynomial approximation of functions on Hilbert space, Math. USSR Sb., 21 (1973), 255–278 0288.41023 CrossrefGoogle Scholar[4] V. Yu. Bentkus and , A. Rachkauskas, Estimates of the rate of convergence of sums of independent random variables in a Banach space. I, Lithuanian Math. J., 22 (1982), 222–234 0522.60007 V. Yu. Bentkus and , A. Rachkauskas, Estimates of the rate of convergence of sums of independent random variables in a Banach space. II, Lithuanian Math. J., 22 (1982), 344–353 0522.60008 CrossrefGoogle Scholar[5] V. Yu. Bentkus and , A. Rachkauskas, On the rate of convergence in the central limit theorem in infinite-dimensional spaces, Lithuanian Math. J., 21 (1981), 271–277 0518.60008 CrossrefGoogle Scholar[6] V. Yu. Bentkus, Estimates of closeness for sums of independent random elements in the space $C[0,1]$, Litovsk. Mat. Sb., 23 (1983), 7–16, (In Russian.) Google Scholar[7] P. P. Gudinas, On the use of metrics of type ζ in limit theorems for weakly dependent random elements, Abstracts, 3rd Internat. Vil'nius Conference on Probability Theory and Mathematical Statistics, Vol. 1, Inst. Matematiki i Kibernetiki Akad. Nauk Litovsk. SSR, Vilnius, 1981, 151–152, (In Russian.) Google Scholar[8] V. V. Senatov, Some lower estimates for the rate of convergence in the central limit theorem in Hilbert space, Soviet Math. Dokl., 23 (1981), 188–192 0474.60007 Google Scholar[9] R. Lapinskas, On the approximation of partial sums in certain Banach spaces, Lithuanian Math. J., 18 (1978), 494–498 0412.60011 CrossrefGoogle Scholar[10] V. M. Zolotarev, Ideal metrics in the problem of approximating the distributions of sums of independent random variables, Theory Prob. Appl., 22 (1977), 433–449. 0385.60025 Google Scholar[11] G. Pisier, Le théorèms de la limits centrâle et la loi de logarithms itéré dans les espaces de Banach, 1975–1976, Sém. Maurey-Schwartz, exp III Google Scholar[12] V. V. Yurinskii, On the error in Gaussian approximation of convolutions, Theory Prob. Appl., 22 (1977), 236–247 LinkGoogle Scholar[13] A. G. Kukush, Weak convergence of measures and convergence of semi-invariants, Theory Probab. Math. Statist., 23 (1981), 79–86 0482.28010 Google Scholar[14] A. G. Kukush, The central limit theorem in Hilbert space in terms of the Lévy-Prokhorov metric, Theory Probab. Math. Statist., 25 (1982), 61–68 0504.60009 Google Scholar[15] R. M. Dudley, Distances of probability measures and random variables, Ann. Math. Statist, 39 (1968), 1563–1572 37:5900 0169.20602 CrossrefGoogle Scholar[16] G. P. Yamukov, Estimates for generalized Dudley metrics in spaces of finite-dimensional distributions, Theory prob. Appl., 22 (1977), 579–584 0386.60022 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On a functional contraction methodThe Annals of Probability, Vol. 43, No. 4 Cross Ref Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements12 November 2012 Cross Ref A functional limit theorem for the profile of search treesThe Annals of Applied Probability, Vol. 18, No. 1 Cross Ref A New Way to Obtain Estimates in the Invariance Principle Cross Ref On the rate of convergence in the martingale CLTStatistics & Probability Letters, Vol. 23, No. 3 Cross Ref Geometric stable distributions in Banach spacesJournal of Theoretical Probability, Vol. 7, No. 2 Cross Ref Rate of Convergence for Sums and Maxima and Doubly Ideal MetricsS. T. Rachev and L. Rüschendorf28 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 2AbstractPDF (1313 KB) Volume 29, Issue 1| 1985Theory of Probability & Its Applications History Submitted:29 October 1981Published online:17 July 2006 InformationCopyright © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1129005Article page range:pp. 50-65ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Central limit theorems for weighted sums of exchangeable random elements in banach spaces(1)

A generalization of a central limit theorem for martingale difference arrays due to D. L. McLeish is obtained for random elements in a separable Banach space E. This result, with a technique of N. C. Weber, is used to obtain a weak convergence theorem for weighted sums of a row-wise exchangeable array (Xnk) of random elements in a Banach space E which is uniformly 2-smooth. Corollaries include a central limit theorem for weighted sums of an exchangeable sequence (Xk), and several weak laws of large numbers for such weighted sums

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

Let {Xk} be independent random variables with EXk = 0 for all k and let {ank : n ≥ 1, k ≥ 1} be an array of real numbers. In this paper the almost sure convergence of , 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.

A Generalization to Dual Banach Spaces of a Theorem by Balakrishnan

A class of optimal control problems is studied in which the controls and outputs are taken as elements in Banach spaces, and the cost functions and constraints are expressible in terms of the norms on these spaces. The paper is principally concerned with generalizing certain results of Balakrishnan relating to optimal control in Hilbert space to this more general setting.