# On the large deviations of a class of modulated additive processes

Ken R. Duffy; Claudio Macci; Giovanni Luca Torrisi

ESAIM: Probability and Statistics (2012)

- Volume: 15, page 83-109
- ISSN: 1292-8100

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topDuffy, Ken R., Macci, Claudio, and Torrisi, Giovanni Luca. "On the large deviations of a class of modulated additive processes." ESAIM: Probability and Statistics 15 (2012): 83-109. <http://eudml.org/doc/222463>.

@article{Duffy2012,

abstract = {
We prove that the large deviation principle holds for a class of
processes inspired by semi-Markov additive processes. For the
processes we consider, the sojourn times in the phase process need
not be independent and identically distributed. Moreover the
state selection process need not be independent of the sojourn
times. We assume that the phase process takes values in a finite set and
that the order in which elements in the set, called states, are
visited is selected stochastically. The sojourn times determine
how long the phase process spends in a state once it has been
selected. The main tool is a representation formula for the sample
paths of the empirical laws of the phase process. Then, based on
assumed joint large deviation behavior of the state selection and
sojourn processes, we prove that the empirical laws of the phase
process satisfy a sample path large deviation principle. From this
large deviation principle, the large deviations behavior of a
class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide
an alternate proof of results for modulated Lévy processes. As a
practical application of the results, we calculate the large deviation
rate function for a processes that arises as the International
Telecommunications Union's standardized stochastic model of two-way
conversational speech.
},

author = {Duffy, Ken R., Macci, Claudio, Torrisi, Giovanni Luca},

journal = {ESAIM: Probability and Statistics},

keywords = {Large deviations; modulated additive processes; speech models; large deviation; modulated additive process; speech model},

language = {eng},

month = {1},

pages = {83-109},

publisher = {EDP Sciences},

title = {On the large deviations of a class of modulated additive processes},

url = {http://eudml.org/doc/222463},

volume = {15},

year = {2012},

}

TY - JOUR

AU - Duffy, Ken R.

AU - Macci, Claudio

AU - Torrisi, Giovanni Luca

TI - On the large deviations of a class of modulated additive processes

JO - ESAIM: Probability and Statistics

DA - 2012/1//

PB - EDP Sciences

VL - 15

SP - 83

EP - 109

AB -
We prove that the large deviation principle holds for a class of
processes inspired by semi-Markov additive processes. For the
processes we consider, the sojourn times in the phase process need
not be independent and identically distributed. Moreover the
state selection process need not be independent of the sojourn
times. We assume that the phase process takes values in a finite set and
that the order in which elements in the set, called states, are
visited is selected stochastically. The sojourn times determine
how long the phase process spends in a state once it has been
selected. The main tool is a representation formula for the sample
paths of the empirical laws of the phase process. Then, based on
assumed joint large deviation behavior of the state selection and
sojourn processes, we prove that the empirical laws of the phase
process satisfy a sample path large deviation principle. From this
large deviation principle, the large deviations behavior of a
class of modulated additive processes is deduced. As an illustration of the utility of the general results, we provide
an alternate proof of results for modulated Lévy processes. As a
practical application of the results, we calculate the large deviation
rate function for a processes that arises as the International
Telecommunications Union's standardized stochastic model of two-way
conversational speech.

LA - eng

KW - Large deviations; modulated additive processes; speech models; large deviation; modulated additive process; speech model

UR - http://eudml.org/doc/222463

ER -

## References

top- 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.
- S. Asmussen and M. Pihlsgård, Loss rates for Lévy processes with two reflecting barriers. Math. Oper. Res.32 (2007) 308–321.
- S. Benaim and P. Friz, Smile asymptotics. II. Models with known moment generating functions. J. Appl. Probab.45 (2008) 16–32.
- P. Billingsley, Convergence of probability measures, Wiley Inter-Science (1999).
- P.T. Brady, A statistical analysis of on-off patterns in 16 conversations. The Bell Systems Technical Journal47 (1968) 73–91.
- L. Breuer, On Markov-additive jump processes. Queueing Syst.40 (2002) 75–91.
- 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. Gebiete24 (1972) 85–93; E. Çinlar, Markov additive processes. I, II, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete24 (1972) 95–121.
- A. Dembo and T. Zajic, Large deviations: from empirical mean and measure to partial sums process. Stochastic Process. Appl.57 (1995) 191–224.
- A. Dembo and O. Zeitouni, Large deviation techniques and applications. Springer (1998).
- J-D. Deuschel and D.W. Stroock, Large deviations. Academic Press (1989).
- J.H. Dshalalow, Characterization of modulated Cox measures on topological spaces. Int. J. Appl. Math. Stat.11 (2007) 21–37.
- J.H. Dshalalow and G. Russell, 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.
- K.R. Duffy and A. Sapozhnikov, The large deviation principle for the on-off Weibull sojourn process. J. Appl. Probab.45 (2008) 107–117.
- A. Ganesh, N. O'Connell and D. Wischik, Big queues, Lecture Notes in Mathematics, Vol. 1838. Springer-Verlag, Berlin (2004).
- A.J. Ganesh and N. O'Connell, A large deviation principle with queueing applications. Stochastics and Stochastic Reports73 (2002) 25–35.
- N. Gantert, Functional Erdős-Renyi laws for semiexponential random variables. Ann. Probab.26 (1998) 1356–1369.
- J. Garcia, An extension of the contraction principle. J. Theoret. Probab.17 (2004) 403–434.
- H. Heffes and D.M. Luncantoni, A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE Journal on Selected Areas in Communications4 (1986) 856–868.
- I. Iscoe, P. Ney and E. Nummelin, Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math.6 (1985) 373–412.
- G. Latouche, M. Remiche and P. Taylor, Transient Markov arrival processes. Ann. Appl. Probab.13 (2003) 628–640.
- H.H. Lee and C.K. Un, A study of on-off characteristics of conversation speech. IEEE Transactions on Communications34 (1986) 630–636.
- T. Lehtonen and H. Nyrhinen, On asymptotically efficient simulation of ruin probabilities in a Markovian environment. Scand. Actuar. J. (1992) 60–75. URIMR1193671 (93h:60144)
- K. Majewski, Single class queueing networks with discrete and fluid customers on the time interval ℝ. Queueing Systems36 (2000) 405–435.
- A.P. Markopoulou, F.A. Tobagi and M.J. Karam, Assessing the quality of voice communications over Internet backbones. IEEE Transactions on Networking11 (2003) 747–760.
- A.A. Mogulskii, Large deviations for trajectories of multi-dimensional random walks. Th. Prob. Appl.21 (1976) 300–315.
- S.V. Nagaev, Large deviations of sums of independent random variables. Ann. Probab.7 (1979) 745–789.
- J. Neveu, Une generalisation des processus à accroissements positifs independants. Abh. Math. Sem. Univ. Hamburg25 (1961) 36–61.
- P. Ney and E. Nummelin, Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 (1987) 561–592.
- P. Ney and E. Nummelin, Markov additive processes II. Large deviations. Ann. Probab.15 (1987) 593–609.
- S. Özekici and R. Soyer, Semi-Markov modulated Poisson process: probabilistic and statistical analysis. Math. Methods Oper. Res.64 (2006) 125–144.
- A. Pacheco and N.U. Prabhu, Markov-additive processes of arrivals, Advances in queueing, Probab. Stochastics Ser., CRC, Boca Raton, FL (1995) 167–194.
- A. Puhalskii, Large deviation analysis of the single server queue. Queueing Systems21 (1995) 5–66.
- A. Puhalskii and W. Whitt, Functional large deviation principles for first-passage-time proc esses. Ann. Appl. Probab.7 (1997) 362–381.
- U. Rieder and N. Bäuerle, Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab.42 (2005) 362–378.
- 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).

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