Asymptotically equivalent sequences of matrices pdf

Asymptotically equivalent sequences of matrices and. Asymptotically equivalent sequences of matrices and hermitian block toeplitz matrices with continuous symbols. In order to define the concept of ae sequences of matrices it is necessary to introduce the following two matrix norms. In this section, the concept of asymptotically lacunary i 2invariant equivalence, the concepts of asymptotically lacunary. 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. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by fast and schoenberg. Let g x be any continuous function defined in the appropriate interval. The sequence has statistical limit, denoted by provided that for every, see. On asymptotically statistical equivalence of sequences of. Patterson 2003, extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. In 2003, patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. This study is based on the concept of asymptotically equivalent sequences of nonsquare matrices.

In this paper, the concept of asymptotically lacunary i 2invariant equivalence, the concepts of asymptotically lacunary. On the application of asymptotically equivalent sequences. The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary wss processes. Pdf for a sequencetosequence transformation a, let rmax. Applications to mimo systems january 2009 ieee transactions on information theory 54. On asymptotically ideal equivalent sequences, journal of the egyptian mathematical. Throughout the paper, \\mathbbn\ is the set of all positive integers, and \\mathbbr\ is the set of all real numbers. 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 fibonacci numbers of asymptotically lacunary of. Then, the concept of asymptotically equivalence has been developed by many other researchers see, 10,11,15,17. 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.

On the application of asymptotically equivalent sequences of. Types of convergence of matrices iowa state university. Aug 22, 2019 patterson presented asymptotically statistically equivalent sequences for nonnegative summability matrices. Patterson 2003, extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative sum ability matrices. In 26, asymptotically lacunary statistical equivalent set sequences was studied.

On the asymptotic equivalence of circulant and toeplitz matrices. Two sequences of n n matrices fa n g and fb n g are said to be asymptotically equivalent if max jja n jj2. 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. In addition, they also presented asymptotically equivalent wijsman sense analogs of theorems in 18 due to patterson and sava. Uniform, nonparametric, nonasymptotic con dence sequences steven r. Recently, the concept of asymptotically equivalent was generalizedbybilgin,kumarandsharma,patterson. A characterization of summability matrices that preserve the asymptotic equivalence of two sequences is given. Research article wijsman orlicz asymptotically ideal. A bodelike integral for discrete linear timeperiodic systems.

This paper presents the notion of asymptotically statistical equivalence, which is a natural combination of asymptotic equivalence, and statistical equivalence for sequences of sets. The concept of asymptotically regular matrices which preserve the asymptotic equivalence of two non. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in 9. Asymptotic equality and inequality computer science. 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. Asymptotically liberating sequences of random unitary. The fibonacci numbers of asymptotically lacunary over. In 1993, marouf presented definitions for asymptotically equivalent and asymptotic regular matrices.

A lowcomplexity coding strategy can be found in 5 for. Also in 20, iasymptotically statistical equivalent and iasymptotically. Further, we will present four theorems that characterize asymptotically iequivalent of multiple. 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. 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. On asymptotically statistical equivalence of sequences of sets.

Fridy and orhan studied lacunary statistical convergence. Index termsblock circulant matrices, block toeplitz bt matrices, channel capacity, frobenius norm, functions of ma trices, hermitian matrices, multiple input. Matrix limits, sequences, and series a matrix handbook. 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. 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. Patterson presented asymptotically statistically equivalent sequences for nonnegative summability matrices. Fasymptotically equivalent of multiple l provided that lim n 1 mn m,nx k,l1,1 d. On asymptotically deferred statistical equivalence of sequences. Generalized asymptotically ilacunary statistical equivalent. Asymptotically equivalent sequences of matrices and capacity of a discretetime gaussian mimo channel with memory abstract.

Asymptotically equivalent sequences of matrices and multivariate arma processes article in ieee transactions on information theory 578. Uniform, nonparametric, nonasymptotic con dence sequences. This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and. Asymptotically double lacunary equivalent sequences in. Asymptotic iequivalence of two number sequences and. 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. I asymptotically lacunary statistical g of double sequences. Asymptotically lacunary statistical equivalent sequence spaces defined by ideal convergence and an orlicz function for some. Asymptotically double lacunary statistically equivalent. Limits sequences asymptotically equivalent sequences series matrix functions matrix exponentials matrix limits, sequences, and series a matrix handbook for statisticians wiley online library. We prove the decomposition theorem for asymptotically iequivalent sequences. Lacunaru asymptotically equivalent sequences selma altunda.

Pdf asymptotically equivalent sequences of matrices and. Asymptotically lacunary i2invariant equivalence ios press. 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. In this article we introduce the notion of asymptotically iequivalent sequences. 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. Subsequently, many authors have shown their interest to solve. October 2016 proyecciones journal of mathematics vol. 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. We find its relations to asymptotically statistical convergence, strong asymptotically equivalence, and strong cesaro asymptotically equivalence for sequences of sets. Almost asymptotically statistical equivalent of double. On asymptotically deferred statistical equivalence of. The fibonacci numbers of asymptotically lacunary over metric. This study is based on the concept of asymptotically equivalent sequences of non square matrices.

Asymptotically lacunary statistical equivalent sequence. Also in 20, iasymptotically statistical equivalent and i. Two nonnegative sequences x x k and y y k are said to be asymptotically equivalent if lim k x k y k 1. Research article asymptotically lacunary equivalent sequences. Some useful results on asymptotically equivalent sequences of matrices. In addition, some relationships were obtained among these new equivalence concepts. On asymptotically statistically equivalent sequences, demonstr. On asymptotically ideal equivalent sequences sciencedirect.