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On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure

  • Brenton Gray (a1), Phil Pollett (a1) and Hanjun Zhang (a1)

Abstract

Let S be a countable set and let Q = (q ij , i, jS) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number μ ≥ 0 and a strictly positive probability measure m = (m j , jS) such that ∑ iS m i q ij = −μ m j , jb. We prove that there exists a Q-process P(t) = (p ij (t), i, jS) for which m is a μ-invariant measure, that is ∑ iS m i p ij (t) = eμ t m j , jS. We illustrate our results with reference to the Kolmogorov ‘K1’ chain and a birth-death process with catastrophes and instantaneous resurrection.

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Copyright

Corresponding author

Postal address: Department of Mathematics, University of Queensland, Queensland 4072, Australia.
∗∗ Email address: bgray@pacificsolutions.com.au
∗∗∗ Email address: pkp@maths.uq.edu.au
∗∗∗∗ Email address: hjz@maths.uq.edu.au

References

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[1] Anderson, W. J. (1991). Continuous-Time Markov Chains. An Applications-Oriented Approach. Springer, New York.
[2] Chen, A. Y. and Renshaw, E. (1990). Markov branching processes with instantaneous immigration. Prob. Theory Relat. Fields 87, 209240.
[3] Chen, A. Y. and Renshaw, E. (1993). Existence and uniqueness criteria for conservative uni-instantaneous denumerable Markov processes. Prob. Theory Relat. Fields 94, 427456.
[4] Chung, K. (1967). Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer, New York.
[5] Kendall, D. G. and Reuter, G. E. H. (1956). Some pathological Markov processes with a denumerable infinity of states and the associated semigroups of operators on l. In Proc. Internat. Congr. Math. 1954, Amsterdam, Vol. 3, Noordhoff, Groningen, pp. 377415.
[6] Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337358.
[7] Kolmogorov, A. N. (1951). On the differentiability of the transition probabilities in stationary Markov processes with a denumerable number of states. Moskov. Gos. Univ. Učenye Zapiski Mate. 148, 5359 (in Russian).
[8] Pollett, P. K. (1986). On the equivalence of μ-invariant measures for the minimal process and its q-matrix. Stoch. Process. Appl. 22, 203221.
[9] Pollett, P. K. (1988). Reversibility, invariance and μ-invariance. Adv. Appl. Prob. 20, 600621.
[10] Pollett, P. K. (1991). On the construction problem for single-exit Markov chains. Bull. Austral. Math. Soc. 43, 439450.
[11] Pollett, P. K. and Vere-Jones, D. (1992). A note on evanescent processes. Austral. J. Statist. 34, 531536.
[12] Reuter, G. E. H. (1959). Denumerable Markov processes. II. J. London Math. Soc. 34, 8191.
[13] Reuter, G. E. H. (1969). Remarks on a Markov chain example of Kolmogorov. Z. Wahrscheinlichkeitsth. 13, 315320.
[14] Tweedie, R. L. (1974). Some ergodic properties of the Feller minimal process. Quart. J. Math. Oxford Ser. 25, 485495.
[15] Vere-Jones, D. (1967). Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22, 361386.
[16] Yang, X. Q. (1990). The Construction Theory of Denumerable Markov Processes. John Wiley, Chichester.
[17] Zhang, H. J., Lin, X. and Hou, Z. T. (2001). Invariant distributions of Q-processes. I. Chinese Ann. Math. Ser. A 22, 323330.
[18] Zhang, H. J., Lin, X. and Hou, Z. T. (2002). Invariant distributions of Q-processes. II. Chinese Ann. Math. Ser. A 23, 361370.

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On the existence of uni-instantaneous Q-processes with a given finite μ-invariant measure

  • Brenton Gray (a1), Phil Pollett (a1) and Hanjun Zhang (a1)

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