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Translated Poisson approximation to equilibrium distributions of Markov population processes. (English) Zbl 1214.60041

The authors consider the behavior of equilibrium distributions of density dependent Markov population processes in which the transition rates depend on the density of individuals. The Kolmogorov distance between the equilibrium distribution is known [A. D. Barbour, Adv. Appl. Probab. 12, 591–614 (1980; Zbl 0434.60084)] to have a bound of order \(O(1/\sqrt{n})\). The authors prove that this normal approximation can be substantially strengthened. The line of reasoning is based on the Stein-Chen method and Dynkin’s formula and leads to the approximating distribution in the form of a translated Poisson distribution with the same variance and almost the same mean. In spite of much stronger metric and weaker assumptions the error bounds are of the same order. The results are illustrated by a simple example in which an immigration birth and death process with birth occurring in groups is considered.

MSC:

60J75 Jump processes (MSC2010)
62E17 Approximations to statistical distributions (nonasymptotic)
60J05 Discrete-time Markov processes on general state spaces
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
92D25 Population dynamics (general)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 0434.60084
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References:

[1] Barbour AD (2003) Equilibrium distributions for Markov population processes. Adv Appl Probab 12:591–614 · Zbl 0434.60084 · doi:10.2307/1426422
[2] Barbour AD, Holst L, Janson S (1992) Poisson approximation. Oxford University Press, Oxford · Zbl 0746.60002
[3] Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley, New York
[4] Hamza K, Klebaner FC (1995) Conditions for integrability of Markov chains. J Appl Probab 32:541–547 · Zbl 0835.60066 · doi:10.2307/3215307
[5] Kurtz TG (1970) Solutions of ordinary differential equations as limits of pure jump Markov processes. J Appl Probab 7:49–58 · Zbl 0191.47301
[6] Kurtz TG (1971) Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J Appl Probab 8:344–356 · Zbl 0219.60060
[7] Kurtz TG (1981) Approximation of population processes. CBMS-NSF regional conference series in applied mathematics vol 36. SIAM, Philadelphia
[8] Röllin A (2005) Approximation of sums of conditionally independent random variables by the translated Poisson distribution. Bernoulli 11:1115–1128 · Zbl 1102.60022 · doi:10.3150/bj/1137421642
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