On the Hierarchical Ergodicity Coefficient
Authors: Yilun Shang
In this paper, we deal with the fundamental concepts and properties of ergodicity coefficients in a hierarchical sense by making use of partition. Moreover, we establish a hierarchial Hajnal’s inequality improving some previous results.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1089411Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 972
 M. Akelbek and S. Kirkland, Coefficients of ergodicity and the scrambling index. Linear Algebra Appl., 430(2009) 1111–1130.
 M. Artzrouni, The local coefficient of ergodicity of a nonnegative matrix. SIAM J. Matrix Anal. Appl., 25(2003) 507–516.
 M. Artzrouni and O. Gavart, Nonlinear matrix iterative processes and generalized coefficient of ergodicity. SIAM Matrix Anal. Appl., 21(2000) 1343–1353.
 R. L. Dobrushin, Central limit theorem for nonstationary Markov chains. I. Theory Probab. Appl., 1(1956) 65–79.
 J. Hajnal, Weak ergodicity in non-homogeneous Markov chains. Proc. Camb. Phil. Soc., 54(1958) 233–246.
 Y. Han, W. Lu, and T. Chen, Cluster consensus in discrete-time networks of multi-agents with adapted inputs. To appear in IEEE Trans. Neural Netw. Learn. Syst.
 D. J. Hartfiel, Nonhomogeneous Matrix Products. World Scientific, New Jersey, 2002.
 I. C. F. Ipsen and T. M. Selee, Ergodicity coefficients defined by vector norms. SIAM J. Matrix Anal. Appl., 32(2011) 153–200.
 A. A. Markov, Extension of the law of large numbers to dependent quantities. Izv. Fiz.-Matem. Obsch. Kazan Univ. 15(1906) 135–156.
 U. Pˇaun, A class of ergodicity coefficients, and applications. Math. Rep. (Bucur.), 4(2002) 225–232.
 U. Pˇaun, New classes of ergodicity coefficients, and applications. Math. Rep. (Bucur.), 6(2004) 141–158.
 U. Pˇaun, Weak and uniform weak Δ-ergodicity for
[Δ]-groupable finite Markov chains. Math. Rep. (Bucur.), 6(2004) 275–293.
 A. Paz, Ergodic theorems for infinite probabilistic tables. Ann. Math. Statist., 41(1970) 539–550.
 E. Seneta, On the historical development of the theory of finite inhomogeneous Markov chains. Proc. Camb. Phil. Soc., 74(1973) 507–513.
 E. Seneta, Explicit forms for ergodicity coefficients and spectrum localization. Linear Algebra Appl., 60(1984) 187–197.
 E. Seneta, Non-negative Matrices and Markov Chains. Springer-Verlag, New York, 2006.
 Y. Shang, Exponential random geometric graph process models for mobile wireless networks. Proc. of the International Conference on Cyber-Enabled Distributed Computing and Knowledge Discovery, Zhangjiajie, 2009, 56–61.
 Y. Shang, Multi-agent coordination in directed moving neighborhood random networks. Chin. Phys. B, 19(2010) 070201.
 Y. Shang, L1 group consensus of multi-agent systems with stochastic inputs under directed interaction topology. Int. J. Control, 86(2013) 1–8.
 J. Shen, A geometric approach to ergodic non-homogeneous Markov chains. In: (Eds. T.-X. He) Wavelet Analysis and Multiresolution Methods, Lecture Notes in Pure and Applied Mathematics, 212(2000) 341–366.