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The Large Deviations of Estimating Rate Functions

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Ken Duffy  and
Anthony P. Metcalfe
Ken Duffy*
Affiliation:
National University of Ireland, Maynooth
Anthony P. Metcalfe*
Affiliation:
Trinity College Dublin
*
∗Postal address: Hamilton Institute, National University of Ireland, Maynooth, County Kildare, Ireland. Email address: ken.duffy@nuim.ie
∗∗Postal address: Department of Pure and Applied Mathematics, Trinity College Dublin, Ireland.
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Abstract

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Given a sequence of bounded random variables that satisfies a well-known mixing condition, it is shown that empirical estimates of the rate function for the partial sums process satisfy the large deviation principle in the space of convex functions equipped with the Attouch-Wets topology. As an application, a large deviation principle for estimating the exponent in the tail of the queue length distribution at a single-server queue with infinite waiting space is proved.

Keywords

Large deviation estimatequeue length tail estimate

MSC classification

Secondary: 60F10: Large deviations 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 267 - 274
Copyright
© Applied Probability Trust 2005 

References

Attouch, H. and Wets, R. J.-B. (1983). A convergence theory for saddle functions. Trans. Amer. Math. Soc. 280, 1–41.Google Scholar
Attouch, H. and Wets, R. J.-B. (1986). Isometries for the Legendre–Fenchel transform. Trans. Amer. Math. Soc. 296, 33–60.CrossRefGoogle Scholar
Beer, G. (1993). Topologies on Closed and Closed Convex Sets (Math. Appl. 268). Kluwer, Dordrecht.Google Scholar
Billingsley, P. (1995). Probability and Measure. John Wiley, New York.Google Scholar
Bryc, W. and Dembo, A. (1996). Large deviations and strong mixing. Ann. Inst. H. Poincaré Prob. Statist. 32, 549–569.Google Scholar
Crosby, S. et al. (1997). Statistical properties of a near-optimal measurement-based CAC algorithm. In Proc. IEEE ATM (Lisbon, 1997), eds Casaca, A. et al., IEEE Press, Piscataway, NJ, pp. 103–112.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38). Springer, New York.Google Scholar
Deuschel, J.-D. and Stroock, D. W. (1989). Large Deviations (Pure Appl. Math. 137). Academic Press, Boston, MA.Google Scholar
Duffield, N. G. (2000). A large deviation analysis of errors in measurement based admission control to buffered and bufferless resources. Queueing Systems Theory Appl. 34, 131–168.Google Scholar
Duffield, N. G. et al. (1995). Entropy of ATM traffic streams: a tool for estimating QoS parameters. IEEE J. Selected Areas Commun. 13, 981–990.Google Scholar
Duffy, K., Lewis, J. T. and Sullivan, W. G. (2003). Logarithmic asymptotics for the supremum of a stochastic process. Ann. Appl. Prob. 13, 430–445.CrossRefGoogle Scholar
Ganesh, A. and O'Connell, N. (1999). An inverse of Sanov's theorem. Statist. Prob. Lett. 42, 201–206.Google Scholar
Ganesh, A., Green, P., O'Connell, N. and Pitts, S. (1998). Bayesian network management. Queueing Systems Theory Appl. 28, 267–282.Google Scholar
Ganesh, A. J. and O'Connell, N. (2002). A large deviation principle with queueing applications. Stoch. Stoch. Reports 73, 25–35.CrossRefGoogle Scholar
Glynn, P. and Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability (J. Appl. Prob. Special Vol. 31A), Applied Probability Trust, Sheffield, pp. 413–430.Google Scholar
Györfi, L. et al. (2000). Distribution-free confidence intervals for measurement of effective bandwidths. J. Appl. Prob. 37, 1–12.Google Scholar
Lewis, J. T., Pfister, C.-E. and Sullivan, W. G. (1995). Entropy, concentration of probability and conditional limit theorems. Markov Process. Relat. Fields 1, 319–386.Google Scholar
Lewis, J. T. et al. (1998). Practical connection admission control for ATM networks based on on-line measurements. Comput. Commun. 21, 1585–1596.CrossRefGoogle Scholar
Loynes, R. M. (1962). The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Puhalskii, A. (2001). Large Deviations and Idempotent Probability (Chapman and Hall/CRC Monogr. Surveys Pure Appl. Math. 119). Chapman and Hall/CRC, Boca Raton, FL.Google Scholar