Long time behaviour for Markovian branching-immigration systems

Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss...
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Autor*in:	Li, Junping [verfasserIn] Cheng, Lan [verfasserIn] Li, Liuyan [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2020

Schlagwörter:	Long time behaviour Markovian branching system Immigration

Übergeordnetes Werk:	Enthalten in: Discrete event dynamic systems - Dordrecht [u.a.] : Springer Science + Business Media B.V, 1991, 31(2020), 1 vom: 02. Juli, Seite 37-57
Übergeordnetes Werk:	volume:31 ; year:2020 ; number:1 ; day:02 ; month:07 ; pages:37-57

Links:	Volltext

DOI / URN:	10.1007/s10626-020-00323-z

Katalog-ID:	SPR043462995

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520			\|a Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric.
650		4	\|a Long time behaviour \|7 (dpeaa)DE-He213
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650		4	\|a Immigration \|7 (dpeaa)DE-He213
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700	1		\|a Li, Liuyan \|e verfasserin \|4 aut
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author_variant	j l jl l c lc l l ll
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bklnumber	31.55 31.00
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allfields	10.1007/s10626-020-00323-z doi (DE-627)SPR043462995 (DE-599)SPRs10626-020-00323-z-e (SPR)s10626-020-00323-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.55 bkl 31.00 bkl Li, Junping verfasserin aut Long time behaviour for Markovian branching-immigration systems 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric. Long time behaviour (dpeaa)DE-He213 Markovian branching system (dpeaa)DE-He213 Immigration (dpeaa)DE-He213 Cheng, Lan verfasserin aut Li, Liuyan verfasserin aut Enthalten in Discrete event dynamic systems Dordrecht [u.a.] : Springer Science + Business Media B.V, 1991 31(2020), 1 vom: 02. Juli, Seite 37-57 (DE-627)308447379 (DE-600)1502222-5 1573-7594 nnns volume:31 year:2020 number:1 day:02 month:07 pages:37-57 https://dx.doi.org/10.1007/s10626-020-00323-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.55 ASE 31.00 ASE AR 31 2020 1 02 07 37-57
spelling	10.1007/s10626-020-00323-z doi (DE-627)SPR043462995 (DE-599)SPRs10626-020-00323-z-e (SPR)s10626-020-00323-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.55 bkl 31.00 bkl Li, Junping verfasserin aut Long time behaviour for Markovian branching-immigration systems 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric. Long time behaviour (dpeaa)DE-He213 Markovian branching system (dpeaa)DE-He213 Immigration (dpeaa)DE-He213 Cheng, Lan verfasserin aut Li, Liuyan verfasserin aut Enthalten in Discrete event dynamic systems Dordrecht [u.a.] : Springer Science + Business Media B.V, 1991 31(2020), 1 vom: 02. Juli, Seite 37-57 (DE-627)308447379 (DE-600)1502222-5 1573-7594 nnns volume:31 year:2020 number:1 day:02 month:07 pages:37-57 https://dx.doi.org/10.1007/s10626-020-00323-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.55 ASE 31.00 ASE AR 31 2020 1 02 07 37-57
allfields_unstemmed	10.1007/s10626-020-00323-z doi (DE-627)SPR043462995 (DE-599)SPRs10626-020-00323-z-e (SPR)s10626-020-00323-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.55 bkl 31.00 bkl Li, Junping verfasserin aut Long time behaviour for Markovian branching-immigration systems 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric. Long time behaviour (dpeaa)DE-He213 Markovian branching system (dpeaa)DE-He213 Immigration (dpeaa)DE-He213 Cheng, Lan verfasserin aut Li, Liuyan verfasserin aut Enthalten in Discrete event dynamic systems Dordrecht [u.a.] : Springer Science + Business Media B.V, 1991 31(2020), 1 vom: 02. Juli, Seite 37-57 (DE-627)308447379 (DE-600)1502222-5 1573-7594 nnns volume:31 year:2020 number:1 day:02 month:07 pages:37-57 https://dx.doi.org/10.1007/s10626-020-00323-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.55 ASE 31.00 ASE AR 31 2020 1 02 07 37-57
allfieldsGer	10.1007/s10626-020-00323-z doi (DE-627)SPR043462995 (DE-599)SPRs10626-020-00323-z-e (SPR)s10626-020-00323-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.55 bkl 31.00 bkl Li, Junping verfasserin aut Long time behaviour for Markovian branching-immigration systems 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric. Long time behaviour (dpeaa)DE-He213 Markovian branching system (dpeaa)DE-He213 Immigration (dpeaa)DE-He213 Cheng, Lan verfasserin aut Li, Liuyan verfasserin aut Enthalten in Discrete event dynamic systems Dordrecht [u.a.] : Springer Science + Business Media B.V, 1991 31(2020), 1 vom: 02. Juli, Seite 37-57 (DE-627)308447379 (DE-600)1502222-5 1573-7594 nnns volume:31 year:2020 number:1 day:02 month:07 pages:37-57 https://dx.doi.org/10.1007/s10626-020-00323-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.55 ASE 31.00 ASE AR 31 2020 1 02 07 37-57
allfieldsSound	10.1007/s10626-020-00323-z doi (DE-627)SPR043462995 (DE-599)SPRs10626-020-00323-z-e (SPR)s10626-020-00323-z-e DE-627 ger DE-627 rakwb eng 510 ASE 31.55 bkl 31.00 bkl Li, Junping verfasserin aut Long time behaviour for Markovian branching-immigration systems 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric. Long time behaviour (dpeaa)DE-He213 Markovian branching system (dpeaa)DE-He213 Immigration (dpeaa)DE-He213 Cheng, Lan verfasserin aut Li, Liuyan verfasserin aut Enthalten in Discrete event dynamic systems Dordrecht [u.a.] : Springer Science + Business Media B.V, 1991 31(2020), 1 vom: 02. Juli, Seite 37-57 (DE-627)308447379 (DE-600)1502222-5 1573-7594 nnns volume:31 year:2020 number:1 day:02 month:07 pages:37-57 https://dx.doi.org/10.1007/s10626-020-00323-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.55 ASE 31.00 ASE AR 31 2020 1 02 07 37-57
language	English
source	Enthalten in Discrete event dynamic systems 31(2020), 1 vom: 02. Juli, Seite 37-57 volume:31 year:2020 number:1 day:02 month:07 pages:37-57
sourceStr	Enthalten in Discrete event dynamic systems 31(2020), 1 vom: 02. Juli, Seite 37-57 volume:31 year:2020 number:1 day:02 month:07 pages:37-57
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Long time behaviour Markovian branching system Immigration
dewey-raw	510
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container_title	Discrete event dynamic systems
authorswithroles_txt_mv	Li, Junping @@aut@@ Cheng, Lan @@aut@@ Li, Liuyan @@aut@@
publishDateDaySort_date	2020-07-02T00:00:00Z
hierarchy_top_id	308447379
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We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. 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author	Li, Junping
spellingShingle	Li, Junping ddc 510 bkl 31.55 bkl 31.00 misc Long time behaviour misc Markovian branching system misc Immigration Long time behaviour for Markovian branching-immigration systems
authorStr	Li, Junping
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topic_title	510 ASE 31.55 bkl 31.00 bkl Long time behaviour for Markovian branching-immigration systems Long time behaviour (dpeaa)DE-He213 Markovian branching system (dpeaa)DE-He213 Immigration (dpeaa)DE-He213
topic	ddc 510 bkl 31.55 bkl 31.00 misc Long time behaviour misc Markovian branching system misc Immigration
topic_unstemmed	ddc 510 bkl 31.55 bkl 31.00 misc Long time behaviour misc Markovian branching system misc Immigration
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title	Long time behaviour for Markovian branching-immigration systems
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title_full	Long time behaviour for Markovian branching-immigration systems
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author_browse	Li, Junping Cheng, Lan Li, Liuyan
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title_sort	long time behaviour for markovian branching-immigration systems
title_auth	Long time behaviour for Markovian branching-immigration systems
abstract	Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric.
abstractGer	Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric.
abstract_unstemmed	Abstract Let {X(t);t ≥ 0} be a continuous-time branching-immigration system with branching rates {bk; k ≥ 0,k≠ 1} and immigration rates {ak; k ≥ 1}. We assume that b0 = 0, $m=:{\sum }_{k=1}^{\infty }kb_{k}<\infty $ and $a=:{\sum }_{k=1}^{\infty }ka_{k}<\infty $. In this paper, we first discuss the martingale property of W(t) = e−mtX(t) − m− 1a(1 − e−mt) and prove that it has a limit W. Furthermore, we show that X(t + s)/X(t) converges to ems in probability, W(t) converges to W in probability and X(t + s)/X(t) converges to ems in probability conditioned on W ≥ α (here α is a positive constant) as $t\rightarrow \infty $. The explicit estimates of the above three convergence rates are obtained under various moment conditions on {bk; k ≥ 0,k≠ 1}. It is shown that the rate of the first one is geometric, while the other two are supergeometric.
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title_short	Long time behaviour for Markovian branching-immigration systems
url	https://dx.doi.org/10.1007/s10626-020-00323-z
remote_bool	true
author2	Cheng, Lan Li, Liuyan
author2Str	Cheng, Lan Li, Liuyan
ppnlink	308447379
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isOA_txt	false
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doi_str	10.1007/s10626-020-00323-z
up_date	2024-07-03T18:50:25.429Z
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Long time behaviour for Markovian branching-immigration systems

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