Arrays Of Random Variables Research Articles

On strong deviation theorems concerning array of random variables with applications

On strong deviation theorems concerning array of random variables with applications

Gain-scheduled state estimation for discrete-time complex networks under bit-rate constraints

In this paper, the gain-scheduled state estimation issue is investigated for a kind of complex networks subject to randomly occurring nonlinearities under bit-rate constraints. An array of random variables is introduced to govern the nonlinearities whose occurring probability is a time-varying but bounded value with known upper and lower bounds. A bit-rate constraint model is established and an encoding–decoding mechanism is proposed, under which an upper bound of the decoding error is acquired. The primary purpose of the issue considered in this paper is to design a gain-scheduled state estimator to obtain an estimate of the network state with an acceptable accuracy according to available output measurements. By means of the stochastic analysis and Lyapunov stability theory, a sufficient condition is provided such that the estimation error dynamics achieve the exponentially mean-square ultimate boundedness. The required estimator gain matrix is parameterized by solving a series of matrix inequalities. A numerical simulation is exploited to show the usefulness of the obtained gain-scheduled state estimator.

On strong deviation theorems concerning array of dependent random sequence

Let be an array of random variables on the probability space Let be another probability measure on and, assume that under the law of ξ is row-wise independent. Let be the sample-divergence rate of with respect to related to ξ. A kind of strong deviation theorems, represented by of the arithmetic mean and the geometric mean of ξ are obtained. Moreover, no conditions are imposed on the joint distribution of ξ.

Conditional mean convergence theorems of conditionally dependent random variables under conditions of integrability

We give the conditionally residual h-integrability with exponent r for an array of random variables and establish the conditional mean convergence of conditionally negatively quadrant dependent and conditionally negative associated random variables under this integrability. These results generalize and improve the known ones.

Complete q th moment convergence for arrays of random variables

Let { X n i , 1 ≤ i ≤ n , n ≥ 1 } Open image in new window be an array of random variables with E X n i = 0 Open image in new window and E | X n i | q 0 Open image in new window, where x + = max { x , 0 } Open image in new window. From these results, we can easily obtain some known results on complete q th moment convergence.

A note on the ASCLT for triangular arrays of random variables with an extension to U-statistics

In this paper, we obtain an almost sure central limit theorem for products of independent sums of positive random variables. An extension of the result gives an ASCLT for the U-statistics.

A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables

In this article, applying moment inequality of negatively dependent (ND) random variables which obtained by Asadian et al., the complete convergence theorem for weighted sums of arrays of rowwise ND random variables is discussed. As a result, the complete convergence theorem for ND arrays of random variables is extended. Our results generalize and improve those on complete convergence theorem previously obtained by Hu et al., Ahmed et al, Volodin, and Sung from the independent and identically distributed case to ND sequences.Mathematical Subject Classification: 62F12.

A Conditional Random Field Approach to Unsupervised Texture Image Segmentation

An unsupervised multiresolution conditional random field (CRF) approach to texture segmentation problems is introduced. This approach involves local and long-range information in the CRF neighbourhood to determine the classes of image blocks. Likemost Markov random field (MRF) approaches, the proposed method treats the image as an array of random variables and attempts to assign an optimal class label to each. While most MRFs involve only local information extracted from a small neighbourhood, our method also allows a few long-range blocks to be involved in the labelling process. This alleviates the problem of assigning different class labels to disjoint regions of the same texture and oversegmentation due to the lack of long-range interaction among the neighbouring and distant blocks. The proposed method requires no a priori knowledge of the number and types of regions/textures.

Explicit Conditions for the Convergence of Point Processes Associated to Stationary Arrays

In this article, we consider a stationary array of random variables (which satisfy some asymptotic independence conditions), and the corresponding sequence of point processes. Our main result identifies some explicit conditions for the convergence of the sequence of point processes in terms of the probabilistic behavior of the variables in the array.

On bootstrapping periodic random arrays with increasing period

The aim of the paper is to determine when the periodic block bootstrap, procedure introduced by Chan et al. (Technometrics 46(2):215–224, 2004), can be applied to arrays of random variables. Formal consistency is obtained under α-mixing or m-dependence conditions together with the assumption that the length of the period tends to infinity. On the other hand, if the period is constant, inconsistency is shown. The performance of periodic block bootstrap is also compared in simulations with moving block bootstrap. It is suggested that for the case of long-period data the first method is more effective and much more stable with respect to the length of the block size.

Limit theorems on locally compact Abelian groups

We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem [4, Satz 1.3.6], since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure cannot have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p -adic integers and the p -adic solenoid. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Mean Convergence Theorems for Weighted Sums of Arrays of Residually h-integrable Random Variables Concerning the Weights under Dependence Assumptions

From the classical notion of uniform integrability of a sequence of random variables, a new concept called residual h-integrability is introduced for an array of random variables, concerning an array of constants, which is weaker than other previous related notions of integrability.

Large Deviations for Symmetrised Empirical Measures

In this paper we prove a Large Deviation Principle for the sequence of symmetrised empirical measures \(\frac{1}{n}\sum_{i=1}^{n}\delta_{(X^{n}_{i},X^{n}_{\sigma_{n}(i)})}\) where σn is a random permutation and ((Xin)1≤i≤n)n≥1 is a triangular array of random variables with suitable properties. As an application we show how this result allows to improve the Large Deviation Principles for symmetrised initial-terminal conditions bridge processes recently established by Adams, Dorlas and Konig.

On Continuity of the Pearson Statistic and Sample Quantiles

Convergence with probability one (in probability) of sequences of the sample quantiles and the Pearson statistic that are formed by columns of N× n arrays of random variables and bivariate random vectors respectively is established, n→ ∞. Two applications for the continuity of the Pearson statistics, when sampling is only possible along a sequence converging to an inaccessible targeting value, are presented

Almost sure limit theorems for random allocations

Almost sure limit theorems are presented for random allocations. A general almost sure limit theorem is proved for arrays of random variables. It is applied to obtain almost sure versions of the central limit theorem for the number of empty boxes when the parameters are in the central domain. Almost sure versions of the Poisson limit theorem in the left domain are also proved.

Condition for the Convergence of Maxima of Random Triangular Arrays

We consider the convergence of the maxima of a triangular array of random variables. Sufficient and necessary conditions are discussed, assuming that the underlying distributions are twice differentiable. Our results extend those known for the iid. case.

Mean Convergence Theorems with or without Random Indices for Randomly Weighted Sums of Random Elements in Rademacher Type p Banach Spaces

Some mean convergence theorems are established for randomly weighted sums of the form ∑ j = 1 k n A nj V nj and ∑ j = 1 T n A nj V nj where {A nj , j ≥ 1, n ≥ 1} is an array of random variables, {V nj , j ≥ 1, n ≥ 1} is an array of mean 0 random elements in a separable real Rademacher type p (1 ≤ p ≤ 2) Banach space, and {k n , n ≥ 1} and {T n , n ≥ 1} are sequences of positive integers and positive integer‐valued random variables, respectively. The results take the form or where 1 ≤ r ≤ p. It is assumed that the array {A nj V nj , j ≥ 1, n ≥ 1} is comprised of rowwise independent random elements and that for all n ≥ 1, A nj and V nj are independent for all j ≥ 1 and T n and {A nj V nj , j ≥ 1} are independent. No conditions are imposed on the joint distributions of the random indices {T n , n ≥ 1}. The sharpness of the results is illustrated by examples.

On stable convergence in the central limit theorem

A stable convergence theorem for arrays of random variables is established. Moreover, interchangeable sequences of random variables provides a good application for the main result.

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.

A central limit theorem for exchangeable random variables on a locally compact abelian group

A central limit theorem is given for uniformly infinitesimal triangular arrays of random variables in which the random variables in each row are exchangeable and take values in a locally compact second countable abelian group. The limiting distribution in the theorem is Gaussian.