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. Recently, the concept of asymptotically equivalent was generalizedbybilgin,kumarandsharma,patterson. Asymptotically lacunary statistical equivalent sequence spaces defined by ideal convergence and an orlicz function for some. Patterson presented asymptotically statistically equivalent sequences for nonnegative summability matrices. Then, the concept of asymptotically equivalence has been developed by many other researchers see, 10,11,15,17. Patterson 2003, extended those concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative sum ability matrices. A bodelike integral for discrete linear timeperiodic systems. Uniform, nonparametric, nonasymptotic con dence sequences.
On asymptotically statistically equivalent sequences, demonstr. Subsequently, many authors have shown their interest to solve. On asymptotically ideal equivalent sequences sciencedirect. In 1993, marouf presented definitions for asymptotically equivalent and asymptotic regular matrices. Asymptotically equivalent sequences of matrices and capacity of a discretetime gaussian mimo channel with memory abstract. The concept of asymptotically regular matrices which preserve the asymptotic equivalence of two non. This paper presents the following definitions which is a natural combination of the definition for asymptotically equivalent, statistically limit, lacunary sequences, and. 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. Fasymptotically equivalent of multiple l provided that lim n 1 mn m,nx k,l1,1 d. Pdf asymptotically equivalent sequences of matrices and. 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. Two nonnegative sequences x x k and y y k are said to be asymptotically equivalent if lim k x k y k 1. On the application of asymptotically equivalent sequences of. Limits sequences asymptotically equivalent sequences series matrix functions matrix exponentials matrix limits, sequences, and series a matrix handbook for statisticians wiley online library.
Also in 20, iasymptotically statistical equivalent and i. October 2016 proyecciones journal of mathematics vol. I asymptotically lacunary statistical g of double sequences. Research article wijsman orlicz asymptotically ideal. Almost asymptotically statistical equivalent of double. Types of convergence of matrices iowa state university. On the asymptotic equivalence of circulant and toeplitz matrices. In order to define the concept of ae sequences of matrices it is necessary to introduce the following two matrix norms. Asymptotically double lacunary equivalent sequences in. Uniform, nonparametric, nonasymptotic con dence sequences steven r. Asymptotically liberating sequences of random unitary. Asymptotically double lacunary statistically equivalent. 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 this article we introduce the notion of asymptotically iequivalent sequences. A characterization of summability matrices that preserve the asymptotic equivalence of two sequences is given. Throughout the paper, \\mathbbn\ is the set of all positive integers, and \\mathbbr\ is the set of all real numbers. The concept of convergence of a sequence of real numbers has been extended to statistical convergence independently by fast and schoenberg.
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. 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. Let g x be any continuous function defined in the appropriate interval. On asymptotically ideal equivalent sequences, journal of the egyptian mathematical. Patterson 2003, extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability 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. Fridy and orhan studied lacunary statistical convergence. Further, we will present four theorems that characterize asymptotically iequivalent of multiple.
Asymptotically lacunary i2invariant equivalence ios press. In addition, they also presented asymptotically equivalent wijsman sense analogs of theorems in 18 due to patterson and sava. Pdf for a sequencetosequence transformation a, let rmax. On asymptotically statistical equivalence of sequences of sets. 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. Marouf presented definitions for asymptotically equivalent sequences and asymptotic regular matrices in 9. Applications to mimo systems january 2009 ieee transactions on information theory 54. In this paper, the concept of asymptotically lacunary i 2invariant equivalence, the concepts of asymptotically lacunary. On the application of asymptotically equivalent sequences.
Two sequences of n n matrices fa n g and fb n g are said to be asymptotically equivalent if max jja n jj2. In this section, the concept of asymptotically lacunary i 2invariant equivalence, the concepts of asymptotically lacunary. Asymptotically equivalent sequences of matrices and hermitian block toeplitz matrices with continuous symbols. Research article asymptotically lacunary equivalent sequences. On asymptotically deferred statistical equivalence of sequences. Asymptotically equivalent sequences of matrices and. 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 equivalent sequences of matrices and multivariate arma processes article in ieee transactions on information theory 578. Pdf asymptotic equivalence of sequences and summability. 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. The fibonacci numbers of asymptotically lacunary over. Asymptotically lacunary statistical equivalent sequence. 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.
This study is based on the concept of asymptotically equivalent sequences of non square matrices. Also in 20, iasymptotically statistical equivalent and iasymptotically. Aug 22, 2019 patterson presented asymptotically statistically equivalent sequences for nonnegative summability matrices. Asymptotic equality and inequality computer science. The sequence has statistical limit, denoted by provided that for every, see. A lowcomplexity coding strategy can be found in 5 for. The fibonacci numbers of asymptotically lacunary over metric. We find its relations to asymptotically statistical convergence, strong asymptotically equivalence, and strong cesaro asymptotically equivalence for sequences of sets. On asymptotically statistical equivalence of sequences of. In 2003, patterson extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices. 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. The fibonacci numbers of asymptotically lacunary of. Generalized asymptotically ilacunary statistical equivalent. The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary wss processes. In addition, some relationships were obtained among these new equivalence concepts. 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. 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. Asymptotic iequivalence of two number sequences and. 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. We prove the decomposition theorem for asymptotically iequivalent sequences. This study is based on the concept of asymptotically equivalent sequences of nonsquare matrices. Research article asymptotically lacunary equivalent.
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