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Strong approximation theorems for density dependent Markov chains

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  • Kurtz, Thomas G.

Abstract

A variety of continuous parameter Markov chains arising in applied probability (e.g. epidemic and chemical reaction models) can be obtained as solutions of equations of the form where l[set membership, variant]Zt, the Y1 are independent Poisson processes, and N is a parameter with a natural interpretation (e.g. total population size or volume of a reacting solution). The corresponding deterministic model, satisfies X(t)=x0+ [integral operator]t0 [summation operator] lf1(X(s))ds Under very general conditions limN-->[infinity]XN(t)=X(t) a.s. The process XN(t) is compared to the diffusion processes given by and Under conditions satisfied by most of the applied probability models, it is shown that XN,ZN and V can be constructed on the same sample space in such a way that and

Suggested Citation

  • Kurtz, Thomas G., 1978. "Strong approximation theorems for density dependent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 6(3), pages 223-240, February.
    • Handle: RePEc:eee:spapps:v:6:y:1978:i:3:p:223-240
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    3. Xue, Xiaofeng, 2021. "Moderate deviations of density-dependent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 49-80.
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    6. He, Yuheng & Xue, Xiaofeng, 2023. "Moderate deviations of hitting times of a family of density-dependent Markov chains," Statistics & Probability Letters, Elsevier, vol. 195(C).
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    9. Keliger, Dániel & Horváth, Illés, 2023. "Accuracy criterion for mean field approximations of Markov processes on hypergraphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    10. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "Queueing Theoretic Approaches to Financial Price Fluctuations," Papers math/0703832, arXiv.org.
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    15. Oluseyi Odubote & Daniel F. Linder, 2021. "Estimating Equations for Density Dependent Markov Jump Processes," Mathematics, MDPI, vol. 9(4), pages 1-16, February.
    16. Guodong Pang & Alexander L. Stolyar, 2016. "A service system with on-demand agent invitations," Queueing Systems: Theory and Applications, Springer, vol. 82(3), pages 259-283, April.
    17. Young Myoung Ko & Natarajan Gautam, 2013. "Critically Loaded Time-Varying Multiserver Queues: Computational Challenges and Approximations," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 285-301, May.
    18. Keliger, Dániel & Horváth, Illés & Takács, Bálint, 2022. "Local-density dependent Markov processes on graphons with epidemiological applications," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 324-352.
    19. Florin Avram & Rim Adenane & David I. Ketcheson, 2021. "A Review of Matrix SIR Arino Epidemic Models," Mathematics, MDPI, vol. 9(13), pages 1-14, June.
    20. Ephraim M. Hanks, 2017. "Modeling Spatial Covariance Using the Limiting Distribution of Spatio-Temporal Random Walks," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(518), pages 497-507, April.
    21. Hodgkinson, Liam & McVinish, Ross & Pollett, Philip K., 2020. "Normal approximations for discrete-time occupancy processes," Stochastic Processes and their Applications, Elsevier, vol. 130(10), pages 6414-6444.
    22. Ramandeep S. Randhawa & Sunil Kumar, 2009. "Multiserver Loss Systems with Subscribers," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 142-179, February.
    23. Lücke, Marvin & Heitzig, Jobst & Koltai, Péter & Molkenthin, Nora & Winkelmann, Stefanie, 2023. "Large population limits of Markov processes on random networks," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    24. Kuang Xu & Se-Young Yun, 2020. "Reinforcement with Fading Memories," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1258-1288, November.
    25. Jamol Pender & Young Myoung Ko, 2017. "Approximations for the Queue Length Distributions of Time-Varying Many-Server Queues," INFORMS Journal on Computing, INFORMS, vol. 29(4), pages 688-704, November.

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