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.
Keywords: 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://archive.numdam.org/articles/10.1051/ps:2004002/} }
TY - JOUR AU - Paroissin, Christian AU - Ycart, Bernard TI - Central limit theorem for hitting times of functionals of Markov jump processes JO - ESAIM: Probability and Statistics PY - 2004 SP - 66 EP - 75 VL - 8 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps:2004002/ DO - 10.1051/ps:2004002 LA - en ID - PS_2004__8__66_0 ER -
%0 Journal Article %A Paroissin, Christian %A Ycart, Bernard %T Central limit theorem for hitting times of functionals of Markov jump processes %J ESAIM: Probability and Statistics %D 2004 %P 66-75 %V 8 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps:2004002/ %R 10.1051/ps:2004002 %G en %F PS_2004__8__66_0
Paroissin, Christian; Ycart, Bernard. Central limit theorem for hitting times of functionals of Markov jump processes. ESAIM: Probability and Statistics, Volume 8 (2004), pp. 66-75. doi : 10.1051/ps:2004002. http://archive.numdam.org/articles/10.1051/ps:2004002/
[1] Applied probability and queues. Wiley, New York (1987). | MR | Zbl
,[2] Matrix-analytic models and their analysis. Scand. J. Statist. 27 (2000) 193-226. | Zbl
,[3] Stochastic models in reliability. Springer, New York (1999). | MR | Zbl
and ,[4] Mathematical theory of reliability. SIAM, Philadelphia (1996). | MR | Zbl
and ,[5] Elements of applied stochastic processes. Wiley, New York (1984). | MR | Zbl
,[6] An automated stopping rule for MCMC convergence assessment. Comput. Statist. 14 (1999) 419-442. | Zbl
and ,[7] Processus stochatisques et fiablité des systèmes. Springer, Paris (1997). | MR
,[8] Real analysis and probability. Chapman and Hall, London (1989). | MR
,[9] Markov processes: characterization and convergence. Wiley, New York (1986). | MR | Zbl
and ,[10] Central limit theorem in . Z. Wahrsch. Verw. Geb 44 (1978) 89-101. | Zbl
,[11] System reliability theory: models and statistical methods. Wiley, New York (1994). | MR | Zbl
and ,[12] Structured stochastic matrices of type and their applications. Dekker, New York (1989). | MR | Zbl
,[13] Matrix-geometric solutions in stochastic models: an algorithmic approach. Dover, New York (1994). | MR
,[14] Reliability analysis of -out-of- systems with partially repairable multi-state components. Microelectron. Reliab. 36 (1996) 1407-1415.
, and ,[15] Convergence of stochastic processes. Springer, New York (1984). | MR | Zbl
,[16] Approximate distributions of order statistics, with application to non-parametric statistics. Springer, New York (1989). | MR | Zbl
,[17] Stochastic models: an algorithmic approach. Wiley, Chichester (1994). | MR | Zbl
,[18] Some useful functions for functional limit theorems. Math. Oper. Res. 5 (1980) 67-85. | Zbl
,[19] Cutoff for samples of Markov chains. ESAIM: PS 3 (1999) 89-107. | Numdam | Zbl
,[20] Stopping tests for Monte-Carlo Markov chain methods. Meth. Comp. Appl. Probab. 2 (2000) 23-36. | Zbl
,[21] Cutoff for Markov chains: some examples and applications. in Complex Systems, E. Goles and S. Martínez Eds., Kluwer, Dordrecht (2001) 261-300.
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