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On occupation time for on-off processes with multiple off-states
Volume 9, Issue 4 (2022), pp. 413–430
Chaoran Hu ORCID icon link to view author Chaoran Hu details   Vladimir Pozdnyakov   Jun Yan  

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https://doi.org/10.15559/22-VMSTA210
Pub. online: 21 June 2022      Type: Research Article      Open accessOpen Access

Received
15 January 2022
Revised
9 April 2022
Accepted
5 June 2022
Published
21 June 2022

Abstract

The need to model a Markov renewal on-off process with multiple off-states arise in many applications such as economics, physics, and engineering. Characterization of the occupation time of one specific off-state marginally or two off-states jointly is crucial to understand such processes. The exact marginal and joint distributions of the off-state occupation times are derived. The theoretical results are confirmed numerically in a simulation study. A special case when all holding times have Lévy distribution is considered for the possibility of simplification of the formulas.

References

[1] 
Bshouty, D., Di Crescenzo, A., Martinucci, B., Zacks, S.: Generalized telegraph process with random delays. J. Appl. Probab. 49, 850–865 (2012). MR3012104. https://doi.org/10.1017/s002190020000958x
[2] 
Cane, V.R.: Behavior sequences as semi-Markov chains. J. R. Stat. Soc., Ser. B 21, 36–58 (1959). MR0109095
[3] 
Crimaldi, I., Di Crescenzo, A., Iuliano, A., Martinucci, B.: A generalized telegraph process with velocity driven by random trials. Adv. Appl. Probab. 45(4), 1111–1136 (2013). MR3161299. https://doi.org/10.1239/aap/1386857860
[4] 
De Gregorio, A., Macci, C.: Large deviation principles for telegraph processes. Stat. Probab. Lett. 82(11), 1874–1882 (2012). MR2970286
[5] 
Di Crescenzo, A.: On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Probab. 33, 690–701 (2001). MR1860096. https://doi.org/10.1239/aap/1005091360
[6] 
Di Crescenzo, A., Iuliano, A., Martinucci, B., Zacks, S.: Generalized telegraph process with random jumps. J. Appl. Probab. 50(2), 450–463 (2013). MR3102492
[7] 
Di Crescenzo, A., Martinucci, B., Zacks, S.: On the geometric brownian motion with alternating trend. Mathematical and statistical methods for actuarial sciences and finance, Eds. Perna, C. and Sibillo, M., 81–85 (2014).
[8] 
Di Crescenzo, A., Zacks, S.: Probability law and flow function of Brownian motion driven by a generalized telegraph process. Methodol. Comput. Appl. Probab. 17(3), 761–780 (2015). MR3377859. https://doi.org/10.1007/s11009-013-9392-1
[9] 
Kolesnik, A.D., Ratanov, N.: Telegraph Processes and Option Pricing. Springer Briefs in Statistics. Springer (2013). MR3115087. https://doi.org/10.1007/978-3-642-40526-6
[10] 
Macci, C.: Large deviations for some non-standard telegraph processes. Stat. Probab. Lett. 110, 119–127 (2016). MR3474745
[11] 
Newman, D.S.: On the probability distribution of a filtered random telegraph signal. Ann. Math. Stat. 39(3), 890–896 (1968). MR0230564. https://doi.org/10.1214/aoms/1177698321
[12] 
Page, E.S.: Theoretical considerations of routine maintenance. Comput. J. 2(4), 199–204 (1960).
[13] 
Perry, D., Stadje, W., Zacks, S.: First-exit times for increasing compound processes. Commun. Stat., Stoch. Models 15(5), 977–992 (1999). MR1721237. https://doi.org/10.1080/15326349908807571
[14] 
Pozdnyakov, V., Elbroch, L.M., Hu, C., Meyer, T., Yan, J.: On estimation for brownian motion governed by telegraph process with multiple off states. Methodol. Comput. Appl. Probab. 22, 1275–1291 (2020). MR4129134. https://doi.org/10.1007/s11009-020-09774-1
[15] 
Pozdnyakov, V., Elbroch, L.M., Labarga, A., Meyer, T., Yan, J.: Discretely observed Brownian motion governed by telegraph process: estimation. Methodol. Comput. Appl. Probab. 21(3), 907–920 (2019). MR4001858. https://doi.org/10.1007/s11009-017-9547-6
[16] 
Ratanov, N.: Piecewise linear process with renewal starting points. Stat. Probab. Lett. 131, 78–86 (2017). MR3706699. https://doi.org/10.1016/j.spl.2017.08.010
[17] 
Stadje, W., Zacks, S.: Telegraph processes with random velocities. J. Appl. Probab. 41, 665–678 (2004). MR2074815. https://doi.org/10.1239/jap/1091543417
[18] 
Xu, Y., De, S.K., Zacks, S.: Exact distribution of intermittently changing positive and negative compound Poisson process driven by an alternating renewal process and related functions. Probab. Eng. Inf. Sci. 29(3), 385–397 (2015). MR3355610
[19] 
Zacks, S.: Generalized integrated telegraph processes and the distribution of related stopping times. J. Appl. Probab. 41(2), 497–507 (2004). MR2052587. https://doi.org/10.1239/jap/1082999081

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Keywords
Lévy distribution Markov renewal process telegraph process

MSC2010
60G05 60G07

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