Asymptotically equivalent sequences of matrices pdf

Patterson presented asymptotically statistically equivalent sequences for nonnegative summability matrices. In 1993, marouf presented definitions for asymptotically equivalent and asymptotic regular matrices. The fibonacci numbers of asymptotically lacunary over metric. Using some recent results on asymptotically equivalent sequences of matrices, we present in this paper, a new derivation of the capacity formula given by brandenburg and wyner for a discretetime gaussian multipleinputmultipleoutput channel with memory. Also in 20, iasymptotically statistical equivalent and iasymptotically. The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary wss processes. In this article we introduce the notion of asymptotically iequivalent sequences. Limits sequences asymptotically equivalent sequences series matrix functions matrix exponentials matrix limits, sequences, and series a matrix handbook for statisticians wiley online library. Recently, the concept of asymptotically equivalent was generalizedbybilgin,kumarandsharma,patterson. In order to define the concept of ae sequences of matrices it is necessary to introduce the following two matrix norms. On asymptotically statistical equivalence of sequences of sets. On asymptotically deferred statistical equivalence of. On asymptotically statistically equivalent sequences, demonstr.

In 2003, patterson extended these concepts by introducing asymptotically statistical equivalent sequences, an analog of the above definitions, and investigated natural regularity conditions for nonnegative summability matrices. The sequence has statistical limit, denoted by provided that for every, see. Asymptotic iequivalence of two number sequences and. In addition, they also presented asymptotically equivalent wijsman sense analogs of theorems in 18 due to patterson and sava. This study is based on the concept of asymptotically equivalent sequences of non square matrices. A bodelike integral for discrete linear timeperiodic systems. Let g x be any continuous function defined in the appropriate interval. Asymptotically equivalent sequences of matrices and capacity of a discretetime gaussian mimo channel with memory abstract.

Research article wijsman orlicz asymptotically ideal. Asymptotically liberating sequences of random unitary. Asymptotically lacunary statistical equivalent sequence. Patterson 2003, extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative sum ability matrices. The two nonnegative double sequences x x mn and y y mn are said to be asymptotically double equivalent of multiple lprovided that p lim mn xmn ymn land simply asymptotically double equivalent if l 1. Matrix limits, sequences, and series a matrix handbook. Asymptotic equality and inequality computer science.

The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by fast and schoenberg. On asymptotically ideal equivalent sequences, journal of the egyptian mathematical. In 2003, patterson extended these concepts by introducing asymptotically statistical equivalent sequences, an analog of the above definitions, and investigated natural regularity conditions for nonnegative summability matrices see. In this section, the concept of asymptotically lacunary i 2invariant equivalence, the concepts of asymptotically lacunary. Aug 22, 2019 patterson presented asymptotically statistically equivalent sequences for nonnegative summability matrices. Asymptotically equivalent sequences of matrices and. Patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these defi nitions and natural regularity conditions for nonnegative summability matrices in 10.

On asymptotically deferred statistical equivalence of sequences. Asymptotically lacunary statistical equivalent sequence spaces defined by ideal convergence and an orlicz function for some. If l 1 in 2, then the sequences x and y are simply called asymptotically statistical equivalent and it is denoted by x. Two sequences of n n matrices fa n g and fb n g are said to be asymptotically equivalent if max jja n jj2. On asymptotically statistical equivalence of sequences of. On the asymptotic equivalence of circulant and toeplitz matrices. Asymptotically lacunary i2invariant equivalence ios press. Almost asymptotically statistical equivalent of double. Research article asymptotically lacunary equivalent sequences. On the application of asymptotically equivalent sequences of. October 2016 proyecciones journal of mathematics vol. We prove the decomposition theorem for asymptotically iequivalent sequences. The fibonacci numbers of asymptotically lacunary over.

A multivariate random process is said to be asymptotically wss if it has constant mean and the sequence of its autocorrelation matrices is asymptotically equivalent a. Subsequently, many authors have shown their interest to solve. In this paper, the most important results of the cited monograph are generalized to block toeplitz bt matrices by maintaining the same mathematical tools used by gray, that is, by using asymptotically equivalent sequences of matrices. Types of convergence of matrices iowa state university. Further, we will present four theorems that characterize asymptotically iequivalent of multiple. This paper presents the notion of asymptotically statistical equivalence, which is a natural combination of asymptotic equivalence, and statistical equivalence for sequences of sets.

In addition, some relationships were obtained among these new equivalence concepts. On asymptotically ideal equivalent sequences sciencedirect. Two nonnegative sequences x x k and y y k are said to be asymptotically equivalent if lim k x k y k 1 and it is denoted by x. Index termsblock circulant matrices, block toeplitz bt matrices, channel capacity, frobenius norm, functions of ma trices, hermitian matrices, multiple input. In this paper, the concept of asymptotically lacunary i 2invariant equivalence, the concepts of asymptotically lacunary.

Asymptotically double lacunary statistically equivalent sequences of interval numbers ayhan esi adiyaman university, turkey shyamal debnath tripura university, india and subrata saha tripura university, india received. I asymptotically lacunary statistical g of double sequences. For the engineering community, grays tutorial monograph on toeplitz and circulant matrices has been, and remains, the best elementary introduction to the szego theory on large toeplitz matrices. Uniform, nonparametric, nonasymptotic con dence sequences. Generalized asymptotically ilacunary statistical equivalent. A lowcomplexity coding strategy can be found in 5 for. Asymptotically equivalent sequences of matrices and hermitian block toeplitz matrices with continuous symbols. Then, the concept of asymptotically equivalence has been developed by many other researchers see, 10,11,15,17.

Also in 20, iasymptotically statistical equivalent and i. A characterization of summability matrices that preserve the asymptotic equivalence of two sequences is given. In 2003, patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in 9. Two nonnegative sequences x x k and y y k are said to be asymptotically equivalent if lim k x k y k 1.

Fasymptotically equivalent of multiple l provided that lim n 1 mn m,nx k,l1,1 d. Asymptotically double lacunary equivalent sequences in. Lacunaru asymptotically equivalent sequences selma altunda. The fibonacci numbers of asymptotically lacunary of. Asymptotically double lacunary statistically equivalent. The goal of this paper is to present two theorems that characterize asymptotically statistical equivalent of multiple l and the regularity of asymptotically statistical convergence by using a sequence of in. Wijsman orlicz asymptotically idealstatistical equivalent sequences inthissection,wede nethenotionofcesaroorliczasymptotically statistical equivalent and orlicz asymptotically ideal statistical equivalent sequences in wijsman sense and establish some interesting relationships between these notions. Patterson 2003, extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. The concept of asymptotically regular matrices which preserve the asymptotic equivalence of two non. Pdf asymptotically equivalent sequences of matrices and.

Pdf for a sequencetosequence transformation a, let rmax. On the application of asymptotically equivalent sequences. Some useful results on asymptotically equivalent sequences of matrices. Fridy and orhan studied lacunary statistical convergence. Applications to mimo systems january 2009 ieee transactions on information theory 54. In 26, asymptotically lacunary statistical equivalent set sequences was studied.

Throughout the paper, \\mathbbn\ is the set of all positive integers, and \\mathbbr\ is the set of all real numbers. Pdf asymptotic equivalence of sequences and summability. Research article asymptotically lacunary equivalent. Asymptotically equivalent sequences of matrices and multivariate arma processes article in ieee transactions on information theory 578.

This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and. This study is based on the concept of asymptotically equivalent sequences of nonsquare matrices. The present monograph studies the asymptotic behaviour of eigenvalues, products and functions of block toeplitz matrices generated by the fourier coefficients of a continuous matrixvalued function. Uniform, nonparametric, nonasymptotic con dence sequences steven r. We find its relations to asymptotically statistical convergence, strong asymptotically equivalence, and strong cesaro asymptotically equivalence for sequences of sets.