Sums Of Independent Random Elements Research Articles

A limit theorem for sums of independent random elements in a Banach space

A limit theorem for sums of independent random elements in a Banach space

On complete convergence in mean for double sums of independent random elements in Banach spaces

For a double array of random elements {T m,n , m ≥ 1, n ≥ 1} in a real separable Banach space X, we study the notion of T m,n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T m,n converging completely to 0 and (ii) T m,n converging to 0 in mean of order p as max{m, n} →∞. When X is of Rademacher type p (1 ≤ p ≤ 2), for a double array of independent mean 0 random elements {V m,n , m ≥ 1, n ≥ 1} in X and a double array of constants {b m,n , m ≥ 1, n ≥ 1}, conditions are provided under which max1≤k≤m,1≤l≤n||Ʃi=1 kƩj=1 l V i,j||/b m,n converges completely to 0 in mean of order p. Moreover, these conditions are shown to provide an exact characterization of Rademacher type p (1 ≤ p ≤ 2) Banach spaces. Illustrative examples are provided.

On complete convergence in mean for double sums of independent random elements in Banach spaces

Conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type p Banach space converges completely to 0 in mean of order p. These conditions for the complete convergence in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. In case the Banach space is not of Rademacher type p, it is proved that the complete convergence in mean of order p of a normed double sum implies a strong law of large numbers.

Almost Sure and Complete Convergence of Randomly Weighted Sums of Independent Random Elements in Banach Spaces

This work develops almost sure and complete convergence of randomly weighted sums of independent random elements in a real separable Banach space. Sufficient conditions for almost sure convergence and complete convergence in the sense defined by Hsu and Robbins are provided. Examples showing that the conditions cannot be removed or weakened are given. It is also demonstrated that some of the known theorems in the literature are special cases of our results

Weak laws of large numbers for double sums of independent random elements in Rademacher type p and stable type p Banach spaces

For a double array { V m n , m ≥ 1 , n ≥ 1 } of independent random elements in a real separable stable type p ( 1 ≤ p < 2 ) Banach space X and sequences of random positive integers { T n , n ≥ 1 } and { τ n , n ≥ 1 } , the main result provides conditions for a weak law of large numbers of the form ∑ i = 1 T m ∑ j = 1 τ n ( V i j − c ( m , n , i , j ) ) / β ( m , n ) → P 0 as max { m , n } → ∞ to hold where the c ( m , n , i , j ) are suitable elements in X and the β ( m , n ) are suitable norming constants. The conditions are shown to completely characterize stable type p ( 1 ≤ p < 2 ) Banach spaces. Illustrative examples are provided. Moreover, for a double array of independent random elements in a real separable Rademacher type p ( 1 ≤ p ≤ 2 ) Banach space, a weak law of large numbers is obtained for the double sums ∑ i = 1 m ∑ j = 1 n V i j , m ≥ 1 , n ≥ 1 .

Strong and Weak Laws of Large Numbers for Double Sums of Independent Random Elements in Rademacher Type p Banach Spaces

For a double array of independent random elements {V mn , m ≥ 1, n ≥ 1} in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space, strong and weak laws of large numbers are established for the double sums , m ≥ 1, n ≥ 1. For the strong law result, it is assumed that EV mn = 0, m ≥ 1, n ≥ 1 and conditions are provided under which almost surely as max {m, n} → ∞. These conditions for the strong law are shown to completely characterize Rademacher type p Banach spaces. The weak law results provide conditions for as max {m, n} → ∞ or for as max {m, n} → ∞ to hold where c ijmn = E(V ij I(‖V ij ‖ ≤ mn)), i, j, m, n ≥ 1 and {T m , m ≥ 1} and {τ n , n ≥ 1} are sequences of positive integer-valued random variables. Illustrative examples are provided.

On Complete Convergence in Mean of Normed Sums of Independent Random Elements in Banach Spaces

For a sequence of random elements {T n , n ≥ 1} in a real separable Banach space 𝒳, we study the notion of T n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T n converging completely to 0 and (ii) T n converging to 0 in mean of order p. When 𝒳 is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements {V n , n ≥ 1} in 𝒳 and a sequence of constants b n → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.

Large-Deviation Probabilities in Banach Spaces

Theorems concerning exact asymptotics of large-deviation probabilities of sums of independent random elements in a Banach space are proved. We consider probabilities of the following events: sums of independed elements belong to balls such that their centers deviate from the origin as the sample size increases. The results are a version of Bentkus' and Rachkauskas' theorems proved for the exteriors of balls centered at the origin. Bibliography: 11 titles.

Probability Inequalities for Generalized L-Statistics

where Xn:1 ≤ · · · ≤ Xn:n are the order statistics based on the sample {Xi; i ≤ n} and hni : R → R, i = 1, . . . , n, are measurable functions. In particular, if hni(y) = cnih(y) and h(y) is monotone then Φn represents the classical L-statistics. Functionals (1) in this general form are called generalized L-statistics. For the first time, these statistics were introduced in [1, 2] where asymptotic expansions for the distributions of these statistics were given in some particular cases. The Fourier analysis of the distributions of Φn is contained in [3]. Note that the integral-type statistics (integral functionals of the empirical distribution function, for example, the Anderson–Darling–Cramer statistics) can be represented as (1), but not as the classical L-statistics (see e.g. [1, 3]). The main purpose of this paper is to obtain upper bounds for the tail probability and moments of Φn. Exponential bounds for the tail probabilities of the classical L-statistics were obtained in [4] by means of approximation of L-statistics by U -statistics with nondegenerate kernels, which makes it possible to reduce the problem to analogous problems for sums of independent real-valued random variables. The approach of the present paper illustrates the capabilities of multivariate analysis: the problems in question are reduced to analogous problems for sums of independent random elements taking values in a functional Banach space. In the previous paper [5], containing some moment inequalities for generalized L-statistics, we suggested an analogous approach using a special property of the order statistics based on a sample from an exponential distribution. In the present paper, to study generalized L-statistics we essentially use the properties of order statistics based on a sample from the (0, 1)-uniform distribution, although we impose no additional restrictions on the sample distribution for the so-called L-statistics with separated kernels which are introduced below. Note also that the term “generalized L-statistics” was introduced in [6] where a generalization of the classical L-statistics theory was considered in a somewhat different aspect related to another construction of order statistics.

Convergence in distribution of almost all sums of independent random elements to a stable law

Convergence in distribution of almost all sums of independent random elements to a stable law

Asymptotic expansions for distributions of sums of independent random elements of a Hilbert space

Asymptotic expansions for distributions of sums of independent random elements of a Hilbert space

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

Estimates of the proximity of sums of independent random elements in the space c[0, 1

For technical reasons it is convenient for use to conduct the investigation in the space C(T) of continuous functions, defined on the unit circle T. We note that to each random element XeC[0,=] corresponds a random element YeC(T), having the characteristics of the element X, for example, one can set Y(t) = X(]tl). Hence the estimates of proximity of sums of independent random elements obtained in the paper are also applicable to the space C[a, b]. Let R I be the real line with addition as group operation, G be the subgroup of the group

A Weak Convergence Theorem for Random Sums of Independent Random Variables

A limit theorem for random sums of independent random elements of $D\lbrack 0, 1\rbrack$ is proved. The theorem extends and simplifies the proof of a result obtained by R. Pyke in [3]. In a personal communication Professor Pyke remarked to me that the method employed is also the right approach to the problem studied in [4], by adequately defining sequences of random elements of $D\lbrack 0, 1 \rbrack$.