Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-27T19:04:50.233Z Has data issue: false hasContentIssue false
  • English
  • Français

How to estimate the rate function of a cumulative process

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
Anthony P. Metcalfe*
Affiliation:
University College Cork
*
∗Postal address: Hamilton Institute, National University of Ireland, Maynooth, County Kildare, Ireland. Email address: ken.duffy@nuim.ie
∗∗Postal address: Boole Centre for Research in Informatics, University College Cork, County Cork, Ireland.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider two sequences of bounded random variables, a value and a timing process, that satisfy the large deviation principle (LDP) with rate function J(⋅,⋅) and whose cumulative process satisfies the LDP with rate function I(⋅). Under mixing conditions, an LDP for estimates of I constructed by transforming an estimate of J is proved. For the case of a cumulative renewal process it is demonstrated that this approach is favourable to a more direct method, as it ensures that the laws of the estimates converge weakly to a Dirac measure at I.

Keywords

Estimating large deviationscumulative processrenewal process

MSC classification

Primary: 60F10: Large deviations
Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1044 - 1052
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.CrossRefGoogle 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. Kluwer, Dordrecht.CrossRefGoogle 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 '97 (Lisbon, June 1997), pp. 103–112.CrossRefGoogle Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviation Techniques and Applications (Appl. Math. (NY) 38), 2nd edn. Springer, New York.CrossRefGoogle 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.CrossRefGoogle Scholar
Duffield, N. G. and Whitt, W. (1998). Large deviations of inversely related processes with non-linear scalings. Ann. Appl. Prob. 8, 995–1026.CrossRefGoogle 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.CrossRefGoogle Scholar
Duffy, K. and Metcalfe, A. P. (2005). The large deviations of estimating rate functions. J. Appl. Prob. 42, 267–274.CrossRefGoogle Scholar
Duffy, K. and Rodgers-Lee, M. (2004). Some useful functions for functional large deviations. Stoch. Stoch. Reports 76, 267–279.CrossRefGoogle Scholar
Eichelsbacher, P. and Ganesh, A. (2002). Bayesian inference for Markov chains. J. Appl. Prob. 39, 91–99.CrossRefGoogle Scholar
Ganesh, A. (1996). Bias correction in effective bandwidth estimation. Performance Evaluation 27, 319–330.CrossRefGoogle Scholar
Ganesh, A. and O'Connell, N. (1999). An inverse of Sanov's theorem. Statist. Prob. Lett. 42, 201–206.CrossRefGoogle Scholar
Ganesh, A., Green, P., O'Connell, N. and Pitts, S. (1998). Bayesian network management. Queueing Systems Theory Appl. 28, 267–282.CrossRefGoogle Scholar
Glynn, P. W. and Whitt, W. (1994). Large deviations behaviour of counting processes and their inverses. Queueing Systems 17, 107–128.CrossRefGoogle Scholar
Glynn, P. W. and Whitt, W. (1994). Logarithmic asymptotics for steady-state tail probabilities in a single-server queue. In Studies in Applied Probability (J. Appl. Prob. Spec. 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.CrossRefGoogle Scholar
Kuczek, T. and Crank, K. N. (1991). A large-deviation result for regenerative processes. J. Theoret. Prob. 4, 551–561.CrossRefGoogle 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
Paschalidis, I. Ch. and Vassilaras, S. (2001). On the estimation of buffer overflow probabilities from measurements. IEEE Trans. Inf. Theory 47, 178–191.CrossRefGoogle Scholar
Puhalskii, A. and Whitt, W. (1997). Functional large deviation principles for first-passage-time processes. Ann. Appl. Prob. 7, 362–381.CrossRefGoogle Scholar
Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press.CrossRefGoogle Scholar
Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Springer, Berlin.CrossRefGoogle Scholar
Russell, R. (1997). The large deviations of random time changes. , Trinity College Dublin.Google Scholar