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Large deviations: From empirical mean and measure to partial sums process

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  • Dembo, Amir
  • Zajic, Tim

Abstract

The large deviation principle is known to hold for the empirical measures (occupation times) of Polish space valued random variables and for the empirical means of Banach space valued random variables under Markov dependence or mixing conditions, and subject to the appropriate exponential tail conditions. It is proved here that these conditions suffice for the large deviation principle to carry over to the partial sums process corresponding to these objects. As demonstrated, this result yields the large deviations of the cost-sampled empirical distribution and is also relevant in studying the buildup of delays in queuing networks.

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  • Dembo, Amir & Zajic, Tim, 1995. "Large deviations: From empirical mean and measure to partial sums process," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 191-224, June.
    • Handle: RePEc:eee:spapps:v:57:y:1995:i:2:p:191-224
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    References listed on IDEAS

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    1. de Acosta, A., 1994. "Large deviations for vector-valued Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 51(1), pages 75-115, June.
    2. Puhalskii, A., 1994. "The method of stochastic exponentials for large deviations," Stochastic Processes and their Applications, Elsevier, vol. 54(1), pages 45-70, November.
    3. Bryc, Wlodzimierz, 1992. "On large deviations for uniformly strong mixing sequences," Stochastic Processes and their Applications, Elsevier, vol. 41(2), pages 191-202, June.
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    Cited by:

    1. Duffy, Ken & Lobunets, Olena & Suhov, Yuri, 2007. "Loss aversion, large deviation preferences and optimal portfolio weights for some classes of return processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 378(2), pages 408-422.
    2. Zajic, Tim, 1996. "Large deviations for a class of recursive algorithms," Statistics & Probability Letters, Elsevier, vol. 26(2), pages 135-140, February.
    3. Dembo, Amir & Zeitouni, Ofer, 1996. "Large deviations for subsampling from individual sequences," Statistics & Probability Letters, Elsevier, vol. 27(3), pages 201-205, April.
    4. Włodzimierz Bryc & Amir Dembo, 1997. "Large Deviations for Quadratic Functionals of Gaussian Processes," Journal of Theoretical Probability, Springer, vol. 10(2), pages 307-332, April.
    5. Dembo, Amir & Zajic, Tim, 1997. "Uniform large and moderate deviations for functional empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 195-211, May.
    6. Vijay G. Subramanian, 2010. "Large Deviations of Max-Weight Scheduling Policies on Convex Rate Regions," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 881-910, November.
    7. Ken Duffy & David Malone, 2008. "Logarithmic asymptotics for a single-server processing distinguishable sources," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 68(3), pages 509-537, December.
    8. Eichelsbacher, Peter, 1998. "Large deviations in partial sums of U-processes in stronger topologies," Statistics & Probability Letters, Elsevier, vol. 38(3), pages 207-214, June.
    9. Xin Guo & Zhao Ruan & Lingjiong Zhu, 2015. "Dynamics of Order Positions and Related Queues in a Limit Order Book," Papers 1505.04810, arXiv.org, revised Oct 2015.
    10. Djellout, Hacène & Guillin, Arnaud & Samoura, Yacouba, 2017. "Estimation of the realized (co-)volatility vector: Large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2926-2960.

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