On the large deviations of a class of modulated additive processes

Ken R. Duffy; Claudio Macci; Giovanni Luca Torrisi

ESAIM: Probability and Statistics (2011)

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

Abstract

top
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.

How to cite

top

Duffy, Ken R., Macci, Claudio, and Torrisi, Giovanni Luca. "On the large deviations of a class of modulated additive processes." ESAIM: Probability and Statistics 15 (2011): 83-109. <http://eudml.org/doc/277149>.

@article{Duffy2011,
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},
pages = {83-109},
publisher = {EDP-Sciences},
title = {On the large deviations of a class of modulated additive processes},
url = {http://eudml.org/doc/277149},
volume = {15},
year = {2011},
}

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
PY - 2011
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/277149
ER -

References

top
  1. [1] S. Asmussen, Risk theory in a Markovian environment. Scand. Actuar. J. (1989) 69–100. Zbl0684.62073MR1035668
  2. [2] S. Asmussen, Ruin probabilities. Advanced Series on Statistical Science & Applied Probability, Vol. 2, World Scientific Publishing Co. Inc., River Edge, NJ (2000). Zbl1247.91080MR1794582
  3. [3] 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. Zbl1029.60001MR1978607
  4. [4] S. Asmussen and M. Pihlsgård, Loss rates for Lévy processes with two reflecting barriers. Math. Oper. Res.32 (2007) 308–321. Zbl05279725MR2324429
  5. [5] S. Benaim and P. Friz, Smile asymptotics. II. Models with known moment generating functions. J. Appl. Probab. 45 (2008) 16–32. Zbl1151.62079MR2409307
  6. [6] P. Billingsley, Convergence of probability measures, Wiley Inter-Science (1999). Zbl0944.60003MR1700749
  7. [7] P.T. Brady, A statistical analysis of on-off patterns in 16 conversations. The Bell Systems Technical Journal47 (1968) 73–91. 
  8. [8] L. Breuer, On Markov-additive jump processes. Queueing Syst.40 (2002) 75–91. Zbl0995.60079MR1893912
  9. [9] N.R. Chaganty, Large deviations for joint distributions and statistical applications. Sankhyā Ser. A59 (1997) 147–166. Zbl0892.60041MR1665683
  10. [10] 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. Zbl0236.60048MR329047
  11. [11] A. Dembo and T. Zajic, Large deviations: from empirical mean and measure to partial sums process. Stochastic Process. Appl.57 (1995) 191–224. Zbl0838.60026MR1334969
  12. [12] A. Dembo and O. Zeitouni, Large deviation techniques and applications. Springer (1998). Zbl0896.60013MR1619036
  13. [13] J-D. Deuschel and D.W. Stroock, Large deviations. Academic Press (1989). Zbl0705.60029MR997938
  14. [14] J.H. Dshalalow, Characterization of modulated Cox measures on topological spaces. Int. J. Appl. Math. Stat.11 (2007) 21–37. Zbl1157.60049MR2374915
  15. [15] 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. Zbl0753.60090MR1169824
  16. [16] K.R. Duffy and A. Sapozhnikov, The large deviation principle for the on-off Weibull sojourn process. J. Appl. Probab.45 (2008) 107–117. Zbl1137.60012MR2409314
  17. [17] A. Ganesh, N. O'Connell and D. Wischik, Big queues, Lecture Notes in Mathematics, Vol. 1838. Springer-Verlag, Berlin (2004). Zbl1044.60001MR2045489
  18. [18] A.J. Ganesh and N. O'Connell, A large deviation principle with queueing applications. Stochastics and Stochastic Reports73 (2002) 25–35. Zbl1006.60021
  19. [19] N. Gantert, Functional Erdős-Renyi laws for semiexponential random variables. Ann. Probab.26 (1998) 1356–1369. Zbl0945.60026MR1640348
  20. [20] J. Garcia, An extension of the contraction principle. J. Theoret. Probab.17 (2004) 403–434. Zbl1082.60043MR2053715
  21. [21] 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. 
  22. [22] I. Iscoe, P. Ney and E. Nummelin, Large deviations of uniformly recurrent Markov additive processes. Adv. in Appl. Math.6 (1985) 373–412. Zbl0602.60034MR826590
  23. [23] G. Latouche, M. Remiche and P. Taylor, Transient Markov arrival processes. Ann. Appl. Probab.13 (2003) 628–640. Zbl1030.60067MR1970280
  24. [24] H.H. Lee and C.K. Un, A study of on-off characteristics of conversation speech. IEEE Transactions on Communications34 (1986) 630–636. 
  25. [25] 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) Zbl0755.62080MR1193671
  26. [26] K. Majewski, Single class queueing networks with discrete and fluid customers on the time interval ℝ. Queueing Systems36 (2000) 405–435. Zbl0969.60095MR1823977
  27. [27] 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. 
  28. [28] A.A. Mogulskii, Large deviations for trajectories of multi-dimensional random walks. Th. Prob. Appl.21 (1976) 300–315. Zbl0366.60031MR420798
  29. [29] S.V. Nagaev, Large deviations of sums of independent random variables. Ann. Probab.7 (1979) 745–789. Zbl0418.60033MR542129
  30. [30] J. Neveu, Une generalisation des processus à accroissements positifs independants. Abh. Math. Sem. Univ. Hamburg25 (1961) 36–61. Zbl0103.36303MR130714
  31. [31] P. Ney and E. Nummelin, Markov additive processes. I. Eigenvalue properties and limit theorems. Ann. Probab. 15 (1987) 561–592. Zbl0625.60027MR885131
  32. [32] P. Ney and E. Nummelin, Markov additive processes II. Large deviations. Ann. Probab. 15 (1987) 593–609. Zbl0625.60028MR885132
  33. [33] S. Özekici and R. Soyer, Semi-Markov modulated Poisson process: probabilistic and statistical analysis. Math. Methods Oper. Res.64 (2006) 125–144. Zbl1106.60071MR2264776
  34. [34] A. Pacheco and N.U. Prabhu, Markov-additive processes of arrivals, Advances in queueing, Probab. Stochastics Ser., CRC, Boca Raton, FL (1995) 167–194. Zbl0845.60090MR1395158
  35. [35] A. Puhalskii, Large deviation analysis of the single server queue. Queueing Systems21 (1995) 5–66. Zbl0849.60083MR1372048
  36. [36] A. Puhalskii and W. Whitt, Functional large deviation principles for first-passage-time proc esses. Ann. Appl. Probab.7 (1997) 362–381. Zbl0885.60023MR1442318
  37. [37] U. Rieder and N. Bäuerle, Portfolio optimization with unobservable Markov-modulated drift process. J. Appl. Probab.42 (2005) 362–378. Zbl1138.93428MR2145482
  38. [38] R.T. Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J. (1970). Zbl0932.90001MR274683
  39. [39] International Telecommunication Union, Recommendation ITU-T P.59, Artificial Conversational Speech (March 1993). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.