Central limit theorem for hitting times of functionals of Markov jump processes
ESAIM: Probability and Statistics, Tome 8 (2004) , pp. 66-75.

A sample of i.i.d. continuous time Markov chains being defined, the sum over each component of a real function of the state is considered. For this functional, a central limit theorem for the first hitting time of a prescribed level is proved. The result extends the classical central limit theorem for order statistics. Various reliability models are presented as examples of applications.

DOI : https://doi.org/10.1051/ps:2004002
Classification : 60F05,  60J25,  60K10
Mots clés : central limit theorem, hitting time, reliability, failure time
@article{PS_2004__8__66_0,
author = {Paroissin, Christian and Ycart, Bernard},
title = {Central limit theorem for hitting times of functionals of Markov jump processes},
journal = {ESAIM: Probability and Statistics},
pages = {66--75},
publisher = {EDP-Sciences},
volume = {8},
year = {2004},
doi = {10.1051/ps:2004002},
mrnumber = {2085606},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2004002/}
}
Paroissin, Christian; Ycart, Bernard. Central limit theorem for hitting times of functionals of Markov jump processes. ESAIM: Probability and Statistics, Tome 8 (2004) , pp. 66-75. doi : 10.1051/ps:2004002. http://www.numdam.org/articles/10.1051/ps:2004002/

[1] S. Asmussen, Applied probability and queues. Wiley, New York (1987). | MR 889893 | Zbl 0624.60098

[2] S. Asmussen, Matrix-analytic models and their analysis. Scand. J. Statist. 27 (2000) 193-226. | Zbl 0959.60085

[3] T. Aven and U. Jensen, Stochastic models in reliability. Springer, New York (1999). | MR 1679540 | Zbl 0926.60006

[4] R.E. Barlow and F. Proschan, Mathematical theory of reliability. SIAM, Philadelphia (1996). | MR 1392947 | Zbl 0874.62111

[5] U.N. Bhat, Elements of applied stochastic processes. Wiley, New York (1984). | MR 322976 | Zbl 0646.60001

[6] D. Chauveau and J. Diébolt, An automated stopping rule for MCMC convergence assessment. Comput. Statist. 14 (1999) 419-442. | Zbl 0947.60018

[7] C. Cocozza-Thivent, Processus stochatisques et fiablité des systèmes. Springer, Paris (1997). | MR 1619554

[8] R.M. Dudley, Real analysis and probability. Chapman and Hall, London (1989). | MR 982264

[9] S.N. Ethier and T.G. Kurtz, Markov processes: characterization and convergence. Wiley, New York (1986). | MR 838085 | Zbl 0592.60049

[10] M.G. Hahn, Central limit theorem in $D\left[0,1\right]$. Z. Wahrsch. Verw. Geb 44 (1978) 89-101. | Zbl 0378.60002

[11] A. Hølyand and M. Rausand, System reliability theory: models and statistical methods. Wiley, New York (1994). | MR 1300407 | Zbl 0846.93001

[12] M.F. Neuts, Structured stochastic matrices of $M/G/1$ type and their applications. Dekker, New York (1989). | MR 1010040 | Zbl 0695.60088

[13] M.F. Neuts, Matrix-geometric solutions in stochastic models: an algorithmic approach. Dover, New York (1994). | MR 1313503

[14] H. Pham, A. Suprasad and R.B. Misra, Reliability analysis of $k$-out-of-$n$ systems with partially repairable multi-state components. Microelectron. Reliab. 36 (1996) 1407-1415.

[15] D. Pollard, Convergence of stochastic processes. Springer, New York (1984). | MR 762984 | Zbl 0544.60045

[16] R.-D. Reiss, Approximate distributions of order statistics, with application to non-parametric statistics. Springer, New York (1989). | MR 988164 | Zbl 0682.62009

[17] H.C. Tijms, Stochastic models: an algorithmic approach. Wiley, Chichester (1994). | MR 1314821 | Zbl 0838.60075

[18] W. Whitt, Some useful functions for functional limit theorems. Math. Oper. Res. 5 (1980) 67-85. | Zbl 0428.60010

[19] B. Ycart, Cutoff for samples of Markov chains. ESAIM: PS 3 (1999) 89-107. | Numdam | Zbl 0932.60077

[20] B. Ycart, Stopping tests for Monte-Carlo Markov chain methods. Meth. Comp. Appl. Probab. 2 (2000) 23-36. | Zbl 0957.60078

[21] B. Ycart, Cutoff for Markov chains: some examples and applications. in Complex Systems, E. Goles and S. Martínez Eds., Kluwer, Dordrecht (2001) 261-300.