Backward Non-Homogeneous Markov Systems: Weak Ergodicity
The foundation of the novel stochastic process Backward Non-Homogeneous Markov system ( B -NHMS) is provided in the present. This process is closely connected with the problem of tendency to consensus in an information exchanging operation. A problem which has attracted more than a few thousands of...
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| Published in | Methodology and computing in applied probability Vol. 27; no. 3; p. 60 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
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Springer US
01.09.2025
Springer Nature B.V |
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| Abstract | The foundation of the novel stochastic process Backward Non-Homogeneous Markov system (
B
-NHMS) is provided in the present. This process is closely connected with the problem of tendency to consensus in an information exchanging operation. A problem which has attracted more than a few thousands of citations in the literature and where backward products of stochastic matrices appear. For forward NHMS with chronological order it is known that weak ergodicity does not necessarily imply strong ergodicity. In a basic Theorem it is proved that in
B
-NHMS with chronological order weak ergodicity always imply strong ergodicity. Later sufficient and necessary conditions are provided for strong ergodicity of
B
-NHMS to converge with geometrical rate uniformly. Next ergodicity of
B
-NHMS is studied when the input process is a non-homogeneous Poisson process. Finally an illustration of some of the main results are given for a manpower system with four grades. |
|---|---|
| AbstractList | The foundation of the novel stochastic process Backward Non-Homogeneous Markov system (
B
-NHMS) is provided in the present. This process is closely connected with the problem of tendency to consensus in an information exchanging operation. A problem which has attracted more than a few thousands of citations in the literature and where backward products of stochastic matrices appear. For forward NHMS with chronological order it is known that weak ergodicity does not necessarily imply strong ergodicity. In a basic Theorem it is proved that in
B
-NHMS with chronological order weak ergodicity always imply strong ergodicity. Later sufficient and necessary conditions are provided for strong ergodicity of
B
-NHMS to converge with geometrical rate uniformly. Next ergodicity of
B
-NHMS is studied when the input process is a non-homogeneous Poisson process. Finally an illustration of some of the main results are given for a manpower system with four grades. The foundation of the novel stochastic process Backward Non-Homogeneous Markov system ( $$\mathcal {B}$$ B -NHMS) is provided in the present. This process is closely connected with the problem of tendency to consensus in an information exchanging operation. A problem which has attracted more than a few thousands of citations in the literature and where backward products of stochastic matrices appear. For forward NHMS with chronological order it is known that weak ergodicity does not necessarily imply strong ergodicity. In a basic Theorem it is proved that in $$\mathcal {B}$$ B -NHMS with chronological order weak ergodicity always imply strong ergodicity. Later sufficient and necessary conditions are provided for strong ergodicity of $$\mathcal {B}$$ B -NHMS to converge with geometrical rate uniformly. Next ergodicity of $$ \mathcal {B}$$ B -NHMS is studied when the input process is a non-homogeneous Poisson process. Finally an illustration of some of the main results are given for a manpower system with four grades. The foundation of the novel stochastic process Backward Non-Homogeneous Markov system (B-NHMS) is provided in the present. This process is closely connected with the problem of tendency to consensus in an information exchanging operation. A problem which has attracted more than a few thousands of citations in the literature and where backward products of stochastic matrices appear. For forward NHMS with chronological order it is known that weak ergodicity does not necessarily imply strong ergodicity. In a basic Theorem it is proved that in B-NHMS with chronological order weak ergodicity always imply strong ergodicity. Later sufficient and necessary conditions are provided for strong ergodicity of B -NHMS to converge with geometrical rate uniformly. Next ergodicity of B-NHMS is studied when the input process is a non-homogeneous Poisson process. Finally an illustration of some of the main results are given for a manpower system with four grades. |
| ArticleNumber | 60 |
| Author | Vassiliou, P.-C.G |
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| Cites_doi | 10.1007/BF01457949 10.1080/03610926.2012.698782 10.2307/3213839 10.2307/3214382 10.1017/S0305004100033399 10.1080/15326349.2014.912947 10.1016/0024-3795(92)90216-W 10.1080/03610926.2012.755200 10.2307/2344779 10.1509/jmkg.73.5.90 10.2307/3212978 10.1017/S0305004100030991 10.1007/BF01900140 10.1137/1101006 10.1287/opre.38.3.537 10.1016/0304-4149(86)90008-6 10.3390/math8061021 10.2307/2344714 10.2307/3213262 10.1080/03610926.2013.788713 10.1057/jors.1976.6 10.2307/3212838 10.1017/S0305004100077276 10.2307/3213497 10.2307/1426955 10.1090/conm/149/01267 10.2307/3214890 10.1007/0-387-32792-4 10.1007/s11009-024-10101-1 10.1007/978-3-030-45982-6 10.2307/3213465 10.2307/3212315 10.1080/01621459.1974.10480137 10.1080/03610926.2012.754468 10.1007/BF00533318 10.3390/math12050660 10.1162/00335530360698432 |
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| References | A Young (10189_CR50) 1974; 138 S Chatterjiee (10189_CR6) 1997; 14 W Doeblin (10189_CR10) 1937; 236 10189_CR45 10189_CR46 E Seneta (10189_CR31) 1979; 11 A Mehlmann (10189_CR24) 1977; 14 P-CG Vassiliou (10189_CR47) 1990; 38 MH De Groot (10189_CR8) 1974; 69 10189_CR4 10189_CR5 P-CG Vassiliou (10189_CR36) 1978; 141 10189_CR34 P-CG Vassiliou (10189_CR49) 1989; 27 P-CG Vassiliou (10189_CR44) 2023 DJ Bartholomew (10189_CR2) 1991 JH Pollard (10189_CR27) 1967; 4 E Seneta (10189_CR33) 2014; 43 SI McClean (10189_CR22) 1980; 17 J Hajnal (10189_CR15) 1956; 52 TA Sarymsakov (10189_CR29) 1958; 120 TA Sarymsakov (10189_CR28) 1956; 8 P-CG Vassiliou (10189_CR42) 2014; 43 SI McClean (10189_CR21) 1976; 13 P-CG Vassiliou (10189_CR35) 1976; 27 SI McClean (10189_CR23) 2014; 43 AC Georgiou (10189_CR14) 1992; 176 J Hajnal (10189_CR17) 1993; 149 JE Cohen (10189_CR7) 1986; 22 P-CG Vassiliou (10189_CR39) 2001 P-CG Vassiliou (10189_CR43) 2020; 8 E Seneta (10189_CR30) 1973; 74 E Duflo (10189_CR11) 2003; 118 AN Kolmogorov (10189_CR20) 1931; 104 C Huang (10189_CR18) 1976; 35 E Seneta (10189_CR32) 1981 SN Bernstein (10189_CR3) 1946 ML Esquivel (10189_CR12) 2014; 30 JD Montgomery (10189_CR25) 1991; 81 P-CG Vassiliou (10189_CR38) 1982; 19 DL Isaacson (10189_CR19) 1976 RL Dobrushin (10189_CR9) 1956; 1 P-CG Vassiliou (10189_CR40) 2010 P-CG Vassiliou (10189_CR48) 1992; 29 P-CG Vassiliou (10189_CR41) 2013; 42 J Hajnal (10189_CR16) 1958; 54 DJ Bartholomew (10189_CR1) 1981 JL Mott (10189_CR26) 1957; 8 P-CG Vassiliou (10189_CR37) 1982; 27 ML Esquivel (10189_CR13) 2017; 15 |
| References_xml | – volume: 104 start-page: 415 year: 1931 ident: 10189_CR20 publication-title: Math Ann doi: 10.1007/BF01457949 – volume: 120 start-page: 465 year: 1958 ident: 10189_CR29 publication-title: Doklady Akad Nauk S.S.S.R. – volume: 42 start-page: 2971 year: 2013 ident: 10189_CR41 publication-title: Commun Stat Theory Meth doi: 10.1080/03610926.2012.698782 – volume: 27 start-page: 851 year: 1982 ident: 10189_CR37 publication-title: J Appl Prob doi: 10.2307/3213839 – ident: 10189_CR4 – volume: 27 start-page: 776 year: 1989 ident: 10189_CR49 publication-title: J Appl Prob doi: 10.2307/3214382 – volume-title: Markov chains Theory and Applications year: 1976 ident: 10189_CR19 – volume: 54 start-page: 233 year: 1958 ident: 10189_CR16 publication-title: Proc Cambridge Philosoph Soc doi: 10.1017/S0305004100033399 – volume: 30 start-page: 365 year: 2014 ident: 10189_CR12 publication-title: Stoch Model doi: 10.1080/15326349.2014.912947 – volume-title: Theoriya Veroiantnostei, 4$$^{th}$$ edition year: 1946 ident: 10189_CR3 – volume: 176 start-page: 137 year: 1992 ident: 10189_CR14 publication-title: Linear Algebra Appl doi: 10.1016/0024-3795(92)90216-W – volume: 43 start-page: 1322 issue: 7 year: 2014 ident: 10189_CR42 publication-title: Commun Stat: Theory Meth doi: 10.1080/03610926.2012.755200 – volume: 141 start-page: 86 year: 1978 ident: 10189_CR36 publication-title: J R Stat Soc A doi: 10.2307/2344779 – ident: 10189_CR34 doi: 10.1509/jmkg.73.5.90 – volume-title: Applied Mathematical Programming year: 2001 ident: 10189_CR39 – volume: 17 start-page: 846 year: 1980 ident: 10189_CR22 publication-title: J Appl Prob doi: 10.2307/3212978 – volume: 52 start-page: 67 year: 1956 ident: 10189_CR15 publication-title: Proc Cambridge Philosoph Soc doi: 10.1017/S0305004100030991 – volume: 8 start-page: 331 year: 1957 ident: 10189_CR26 publication-title: Archiv der Mathematik doi: 10.1007/BF01900140 – volume: 1 start-page: 65 year: 1956 ident: 10189_CR9 publication-title: II Theory Prob Appl doi: 10.1137/1101006 – volume: 38 start-page: 537 issue: 5 year: 1990 ident: 10189_CR47 publication-title: Oper Res doi: 10.1287/opre.38.3.537 – volume: 22 start-page: 315 year: 1986 ident: 10189_CR7 publication-title: Stochast Process Appl doi: 10.1016/0304-4149(86)90008-6 – volume: 8 start-page: 1021 issue: 6 year: 2020 ident: 10189_CR43 publication-title: Mathematics doi: 10.3390/math8061021 – volume-title: Stochastic Models for Social Processes year: 1981 ident: 10189_CR1 – volume: 138 start-page: 584 year: 1974 ident: 10189_CR50 publication-title: J R Stat Soc A doi: 10.2307/2344714 – volume: 14 start-page: 89 year: 1997 ident: 10189_CR6 publication-title: J Appl Probab doi: 10.2307/3213262 – volume: 43 start-page: 1437 issue: 7 year: 2014 ident: 10189_CR23 publication-title: Commun Stat Theory Meth doi: 10.1080/03610926.2013.788713 – volume-title: Statistical Techniques for Manpower Planning year: 1991 ident: 10189_CR2 – volume: 8 start-page: 3 year: 1956 ident: 10189_CR28 publication-title: Doklady Akad Nauk S.S.S.R. – volume: 27 start-page: 57 year: 1976 ident: 10189_CR35 publication-title: Oper Res Quart doi: 10.1057/jors.1976.6 – volume-title: Non-Homogeneous Markov Chains and Systems year: 2023 ident: 10189_CR44 – volume: 13 start-page: 348 year: 1976 ident: 10189_CR21 publication-title: J Appl Prob doi: 10.2307/3212838 – volume: 74 start-page: 507 year: 1973 ident: 10189_CR30 publication-title: Proc Camb Phil Soc doi: 10.1017/S0305004100077276 – volume: 19 start-page: 433 year: 1982 ident: 10189_CR38 publication-title: J Appl Prob doi: 10.2307/3213497 – volume: 15 start-page: 277 year: 2017 ident: 10189_CR13 publication-title: REVSTAT-Stat J – volume: 11 start-page: 576 year: 1979 ident: 10189_CR31 publication-title: Adv Appl Prob doi: 10.2307/1426955 – volume-title: Discrete-time Asset Pricing models in Applied Stochastic Finance year: 2010 ident: 10189_CR40 – volume: 149 start-page: 271 year: 1993 ident: 10189_CR17 publication-title: Contempor Math doi: 10.1090/conm/149/01267 – volume: 81 start-page: 1407 year: 1991 ident: 10189_CR25 publication-title: Am Econ Rev – volume: 29 start-page: 519 issue: 3 year: 1992 ident: 10189_CR48 publication-title: J Appl Prob doi: 10.2307/3214890 – volume: 236 start-page: 3 year: 1937 ident: 10189_CR10 publication-title: Publ Fac Sci Univ Masaryk – volume-title: Non-negative Matrices and Markov Chains year: 1981 ident: 10189_CR32 doi: 10.1007/0-387-32792-4 – ident: 10189_CR46 doi: 10.1007/s11009-024-10101-1 – ident: 10189_CR5 doi: 10.1007/978-3-030-45982-6 – volume: 14 start-page: 611 year: 1977 ident: 10189_CR24 publication-title: J Appl Prob doi: 10.2307/3213465 – volume: 4 start-page: 209 year: 1967 ident: 10189_CR27 publication-title: J Appl Prob doi: 10.2307/3212315 – volume: 69 start-page: 118 issue: 345 year: 1974 ident: 10189_CR8 publication-title: J Am Stat Assoc doi: 10.1080/01621459.1974.10480137 – volume: 43 start-page: 1296 issue: 7 year: 2014 ident: 10189_CR33 publication-title: Commun Stat Theory Methods doi: 10.1080/03610926.2012.754468 – volume: 35 start-page: 141 year: 1976 ident: 10189_CR18 publication-title: Zeitsh Wahrsch and Gebiete doi: 10.1007/BF00533318 – ident: 10189_CR45 doi: 10.3390/math12050660 – volume: 118 start-page: 815 year: 2003 ident: 10189_CR11 publication-title: Quart J Econ doi: 10.1162/00335530360698432 |
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| Snippet | The foundation of the novel stochastic process Backward Non-Homogeneous Markov system (
B
-NHMS) is provided in the present. This process is closely connected... The foundation of the novel stochastic process Backward Non-Homogeneous Markov system ( $$\mathcal {B}$$ B -NHMS) is provided in the present. This process is... The foundation of the novel stochastic process Backward Non-Homogeneous Markov system (B-NHMS) is provided in the present. This process is closely connected... |
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| Title | Backward Non-Homogeneous Markov Systems: Weak Ergodicity |
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