High order moments of first hitting times for single death processes

Abstract

We present an explicit and recursive representation for high order moments of the first hitting times of single death processes. Based on that, some necessary or sufficient conditions of exponential ergodicity as well as a criterion on\mathit{\ell }-ergodicity are obtained for single death processes, respectively.

Keywords

Single death process / moment of the first hitting time / exponential ergodocity

References

[1] Asmussen S, Hering H. Branching Processes. Basel: Birkhauser, 1983

[2] Athreya K B, Ney P E. Branching Processes. Berlin: Springer, 1972

[3] 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

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

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

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

[7] 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)

[8] Wang Z K. Introduction to Stochastic Processes, Vol 2. Beijing: Beijing Normal Univ Press, 1996 (in Chinese)

[9] Zhang Y H. Criteria on ergodicity and strong ergodicity of single death processes. Front Math China, 2018, 13(5): 1215–1243

[10] Zhang Y H. Moments of first hitting times for birth-death processes on trees. Front Math China, 2019, 14(4): 833–854

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature