Criteria on ergodicity and strong ergodicity of single death processes

Abstract

Based on an explicit representation of moments of hitting times for single death processes, the criteria on ergodicity and strong ergodicity are obtained. These results can be applied for an extended class of branching processes. Meanwhile, some sufficient and necessary conditions for recurrence and exponential ergodicity as well as extinction probability for the processes are presented.

Keywords

Single death process / ergodicity / strong ergodicity / recurrence / moments of hitting times

References

[1] Chen A Y, Pollett P, Zhang H J, Cairns B. Uniqueness criteria for continuous-time Markov chains with general transition structure. Adv Appl Probab, 2005, 37(4): 1056–1074

[2] Chen M F. Single birth processes. Chin Ann Math Ser B, 1999, 20: 77–82

[3] Chen M F. Explicit bounds of the first eigenvalue. Sci China Ser A, 2000, 43(10): 1051–1059

[4] Chen M F. Explicit criteria for several types of ergodicity. Chinese J Appl Probab Statist, 2001, 17(2): 1–8

[5] Chen M F. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore: World Scientific, 2004

[6] Chen M F, Zhang Y H. Unified representation of formulas for single birth processes. Front Math China, 2014, 9(4): 761–796

[7] Chen R R. An extended class of time-continuous branching processes. J Appl Probab, 1997, 34(1): 14–23

[8] Hou Z T, Guo Q F. Homogeneous Denumerable Markov Processes. Beijing: Science Press, 1978 (in Chinese); English translation, Beijing: Science Press and Springer, 1988

[9] Isaacson D, Arnold B. Strong ergodicity for continuous-time Markov chains. J Appl Probab, 1978, 15, 699–706

[10] Mao Y H. Ergodic degrees for continuous-time Markov chains. Sci China Math, 2004, 47(2): 161–174

[11] Mao Y H, Zhang Y H. Exponential ergodicity for single-birth processes. J Appl Probab, 2004, 41: 1022–1032

[12] Martínez S, Martín J S, Villemonais D. Existence and uniqueness of a quasi-stationary distribution for Markov processes with fast return from infinity. J Appl Probab, 2014, 51(3): 756–768

[13] Wang L D, Zhang Y H. Criteria for zero-exit (-entrance) of single-birth (-death) Q-matrices. Acta Math Sinica (Chin Ser), 2014, 57(4): 681–692 (in Chinese)

[14] Wang Z K, Yang X Q. Birth and Death Processes and Markov Chains. Science Press: Beijing, 1992

[15] Yan S J, Chen M F. Multidimensional Q-processes. Chin Ann Math Ser B, 1986, 7: 90–110

[16] Yan Y Y, Zhang Y H. Probabilistic meanings for some numerical characteristics of single death processes. J Beijing Normal Univ (Natur Sci), 2018 (to appear) (in Chinese)

[17] Zhang J K. On the generalized birth and death processes (I). Acta Math Sci, 1984, 4: 241–259

[18] Zhang Y H. Strong ergodicity for single-birth processes. J Appl Probab, 2001, 38(1): 270–277

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