Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-15T08:35:57.111Z Has data issue: false hasContentIssue false
  • English
  • Français

On the large deviations of a class of modulated additiveprocesses

Published online by Cambridge University Press:  05 January 2012

Ken R. Duffy,
Claudio Macci and
Giovanni Luca Torrisi
Ken R. Duffy
Affiliation:
Hamilton Institute, National University of Ireland, Maynooth, Ireland; ken.duffy@nuim.ie
Claudio Macci
Affiliation:
Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome, Italy; macci@mat.uniroma2.it
Giovanni Luca Torrisi
Affiliation:
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche (CNR), Via dei Taurini 19, 00185 Rome, Italy; torrisi@iac.rm.cnr.it

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.

We prove that the large deviation principle holds for a class ofprocesses inspired by semi-Markov additive processes. For theprocesses we consider, the sojourn times in the phase process neednot be independent and identically distributed. Moreover thestate selection process need not be independent of the sojourntimes. We assume that the phase process takes values in a finite set andthat the order in which elements in the set, called states, arevisited is selected stochastically. The sojourn times determinehow long the phase process spends in a state once it has beenselected. The main tool is a representation formula for the samplepaths of the empirical laws of the phase process. Then, based onassumed joint large deviation behavior of the state selection andsojourn processes, we prove that the empirical laws of the phaseprocess satisfy a sample path large deviation principle. From thislarge deviation principle, the large deviations behavior of aclass of modulated additive processes is deduced. As an illustration of the utility of the general results, we providean alternate proof of results for modulated Lévy processes. As apractical application of the results, we calculate the large deviationrate function for a processes that arises as the InternationalTelecommunications Union's standardized stochastic model of two-wayconversational speech.

Keywords

Large deviationsmodulated additive processesspeech models

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011, pp. 83 - 109
Copyright
© EDP Sciences, SMAI, 2011

References

S. Asmussen, Risk theory in a Markovian environment. Scand. Actuar. J. (1989) 69–100.
S. Asmussen, Ruin probabilities. Advanced Series on Statistical Science & Applied Probability, Vol. 2, World Scientific Publishing Co. Inc., River Edge, NJ (2000).
S. Asmussen, Applied probability and queues, second edn., Applications of Mathematics (New York), Vol. 51, Springer-Verlag, New York (2003), Stochastic Modelling and Applied Probability.
Asmussen, S. and Pihlsgård, M., Loss rates for Lévy processes with two reflecting barriers. Math. Oper. Res. 32 (2007) 308–321. CrossRef
Benaim, S. and Friz, P., Smile asymptotics. II. Models with known moment generating functions. J. Appl. Probab. 45 (2008) 16–32. CrossRef
P. Billingsley, Convergence of probability measures, Wiley Inter-Science (1999).
Brady, P.T., A statistical analysis of on-off patterns in 16 conversations. The Bell Systems Technical Journal 47 (1968) 73–91. CrossRef
Breuer, L., Markov-additive, On jump processes. Queueing Syst. 40 (2002) 75–91. CrossRef
N.R. Chaganty, Large deviations for joint distributions and statistical applications. Sankhyā Ser. A 59 (1997) 147–166.
E. Çinlar, Markov additive processes. I, II, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 24 (1972) 85–93; E. Çinlar, Markov additive processes. I, II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 (1972) 95–121. CrossRef
Dembo, A. and Zajic, T., Large deviations: from empirical mean and measure to partial sums process. Stochastic Process. Appl. 57 (1995) 191–224. CrossRef
A. Dembo and O. Zeitouni, Large deviation techniques and applications. Springer (1998).
J-D. Deuschel and D.W. Stroock, Large deviations. Academic Press (1989).
Dshalalow, J.H., Characterization of modulated Cox measures on topological spaces. Int. J. Appl. Math. Stat. 11 (2007) 21–37.
Dshalalow, J.H. and Russell, G., On a single-server queue with fixed accumulation level, state dependent service, and semi-Markov modulated input flow. Internat. J. Math. Math. Sci. 15 (1992) 593–600. CrossRef
Duffy, K.R. and Sapozhnikov, A., The large deviation principle for the on-off Weibull sojourn process. J. Appl. Probab. 45 (2008) 107–117. CrossRef
A. Ganesh, N. O'Connell and D. Wischik, Big queues, Lecture Notes in Mathematics, Vol. 1838. Springer-Verlag, Berlin (2004).
Ganesh, A.J. and O'Connell, N., A large deviation principle with queueing applications. Stochastics and Stochastic Reports 73 (2002) 25–35. CrossRef
Gantert, N., Functional Erdős-Renyi laws for semiexponential random variables. Ann. Probab. 26 (1998) 1356–1369.
Garcia, J., An extension of the contraction principle. J. Theoret. Probab. 17 (2004) 403–434. CrossRef
Heffes, H. and Luncantoni, D.M., Markov, A modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE Journal on Selected Areas in Communications 4 (1986) 856–868. CrossRef
Iscoe, I., Ney, P. and Nummelin, E., Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math. 6 (1985) 373–412. CrossRef
Latouche, G., Remiche, M. and Taylor, P., Transient Markov arrival processes. Ann. Appl. Probab. 13 (2003) 628–640.
Lee, H.H. and Un, C.K., A study of on-off characteristics of conversation speech. IEEE Transactions on Communications 34 (1986) 630–636. CrossRef
T. Lehtonen and H. Nyrhinen, On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuar. J. (1992) 60–75. MR1193671 (93h:60144)
Majewski, K., Single class queueing networks with discrete and fluid customers on the time interval ℝ. Queueing Systems 36 (2000) 405–435. CrossRef
Markopoulou, A.P., Tobagi, F.A. and Karam, M.J., Assessing the quality of voice communications over Internet backbones. IEEE Transactions on Networking 11 (2003) 747–760. CrossRef
Mogulskii, A.A., Large deviations for trajectories of multi-dimensional random walks. Th. Prob. Appl. 21 (1976) 300–315. CrossRef
Nagaev, S.V., Large deviations of sums of independent random variables. Ann. Probab. 7 (1979) 745–789. CrossRef
Neveu, J., Une generalisation des processus à accroissements positifs independants. Abh. Math. Sem. Univ. Hamburg 25 (1961) 36–61. CrossRef
P. Ney and E. Nummelin, Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 (1987) 561–592.
Ney, P. and Nummelin, E., Markov additive processes II. Large deviations. Ann. Probab. 15 (1987) 593–609. CrossRef
Özekici, S. and Soyer, R., Semi-Markov modulated Poisson process: probabilistic and statistical analysis. Math. Methods Oper. Res. 64 (2006) 125–144. CrossRef
A. Pacheco and N.U. Prabhu, Markov-additive processes of arrivals, Advances in queueing, Probab. Stochastics Ser., CRC, Boca Raton, FL (1995) 167–194.
Puhalskii, A., Large deviation analysis of the single server queue. Queueing Systems 21 (1995) 5–66. CrossRef
Puhalskii, A. and Whitt, W., Functional large deviation principles for first-passage-time proc esses. Ann. Appl. Probab. 7 (1997) 362–381.
Rieder, U. and Bäuerle, N., Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab. 42 (2005) 362–378. CrossRef
R.T. Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J. (1970).
International Telecommunication Union, Recommendation ITU-T P.59, Artificial Conversational Speech (March 1993).