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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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    Cited by:

    1. Jamaal Ahmad & Mogens Bladt, 2022. "Phase-type representations of stochastic interest rates with applications to life insurance," Papers 2207.11292, arXiv.org, revised Nov 2022.
    2. Xue, Xiaofeng, 2021. "Moderate deviations of density-dependent Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 49-80.
    3. Achal Bassamboo & J. Michael Harrison & Assaf Zeevi, 2006. "Design and Control of a Large Call Center: Asymptotic Analysis of an LP-Based Method," Operations Research, INFORMS, vol. 54(3), pages 419-435, June.
    4. 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).
    5. Jiaqi Zhou & Ilya O. Ryzhov, 2021. "Equilibrium analysis of observable express service with customer choice," Queueing Systems: Theory and Applications, Springer, vol. 99(3), pages 243-281, December.
    6. Nicolas Gast & Bruno Gaujal & Chen Yan, 2023. "Exponential asymptotic optimality of Whittle index policy," Queueing Systems: Theory and Applications, Springer, vol. 104(1), pages 107-150, June.
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    8. Horst, Ulrich, 2010. "Dynamic systems of social interactions," Journal of Economic Behavior & Organization, Elsevier, vol. 73(2), pages 158-170, February.
    9. Jamol Pender & Richard Rand & Elizabeth Wesson, 2020. "A Stochastic Analysis of Queues with Customer Choice and Delayed Information," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1104-1126, August.
    10. Sun, Bo & Sun, Xu & Tsang, Danny H.K. & Whitt, Ward, 2019. "Optimal battery purchasing and charging strategy at electric vehicle battery swap stations," European Journal of Operational Research, Elsevier, vol. 279(2), pages 524-539.
    11. Ankit Gupta & Mustafa Khammash, 2022. "Frequency spectra and the color of cellular noise," Nature Communications, Nature, vol. 13(1), pages 1-18, December.
    12. 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.
    13. 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.
    14. 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.
    15. Ramandeep S. Randhawa & Sunil Kumar, 2009. "Multiserver Loss Systems with Subscribers," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 142-179, February.
    16. Kuang Xu & Se-Young Yun, 2020. "Reinforcement with Fading Memories," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1258-1288, November.
    17. Natiello, Mario A. & Solari, Hernán G., 2020. "Modelling population dynamics based on experimental trials with genetically modified (RIDL) mosquitoes," Ecological Modelling, Elsevier, vol. 424(C).
    18. 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).
    19. Erhan Bayraktar & Ulrich Horst & Ronnie Sircar, 2007. "Queueing Theoretic Approaches to Financial Price Fluctuations," Papers math/0703832, arXiv.org.
    20. Oluseyi Odubote & Daniel F. Linder, 2021. "Estimating Equations for Density Dependent Markov Jump Processes," Mathematics, MDPI, vol. 9(4), pages 1-16, February.
    21. 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.
    22. 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.
    23. 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.
    24. 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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