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Central limit theorems for multivariate semi-Markov sequences and processes, with applications

Part of: Limit theorems Special processes Physiological, cellular and medical topics

Published online by Cambridge University Press:  14 July 2016

Frank Ball
Frank Ball*
Affiliation:
University of Nottingham
*
∗Postal address: School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Email address: fgb@maths.nott.ac.uk
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Abstract

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

Keywords

Aggregated processmultivariate central limit theoremmultivariate semi-Markov processmultivariate semi-Markov sequencereward processsingle ion channel modellingstationary processtime interval omission

MSC classification

Primary: 60K15: Markov renewal processes, semi-Markov processes
Secondary: 60F05: Central limit and other weak theorems 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) 92C05: Biophysics
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 2 , June 1999 , pp. 415 - 432
Copyright
Copyright © Applied Probability Trust 1999 

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References

Ball, F. G. (1997). Empirical clustering of bursts of openings in Markov and semi-Markov models of single channel gating incorporating time interval omission. Adv. Appl. Prob. 29, 909–946.Google Scholar
Ball, F. G., and Milne, R. K. Simple derivations of properties of counting processes associated with Markov renewal processes. Submitted.Google Scholar
Ball, F. G., and Rice, J. A. (1992). Stochastic models for ion channels: introduction and bibliography. Math. Biosci. 112, 189–206.Google Scholar
Ball, F. G., and Sansom, M. S. P. (1988). Aggregated Markov processes incorporating time interval omission. Adv. Appl. Prob. 20, 546–572.Google Scholar
Ball, F. G., Milne, R. K., and Yeo, G. F. (1991). Aggregated semi-Markov processes incorporating time interval omission. Adv. Appl. Prob. 23, 772–797.Google Scholar
Bates, S. E., Sansom, M. S. P., Ball, F. G., Ramsey, R. L., and Usherwood, P. N. R. (1990). Glutamate receptor–channel gating: maximum likelihood analysis of gigaohm seal recordings from locust muscle. Biophys. J. 58, 219–229.Google Scholar
Bickel, P. J. and Ritov, Y. (1996). Inference in hidden Markov models I: Local asymptotic normality in the stationary case. Bernoulli 2, 199–228.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
Csenki, A. (1991). Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes. J. Appl. Prob. 28, 822–832.Google Scholar
Csenki, A. (1995). The number of visits to a subset of the state space by a discrete-parameter semi-Markov process. Statist. Prob. Lett. 22, 71–77.Google Scholar
Dabrowski, A. R., and McDonald, D. (1992). Statistical analysis of multiple ion channel data. Ann. Statist. 20, 1180–1202.Google Scholar
Hawkes, A. G., Jalali, A., and Colquhoun, D. (1990). The distributions of the apparent open times and shut times in a single channel record when brief events cannot be detected. Phil. Trans. R. Soc. London. A 332, 511–538.Google Scholar
Hunter, J. J. (1969). On the moments of Markov renewal processes. Adv. Appl. Prob. 1, 188–210.Google Scholar
Jalali, A., and Hawkes, A. G. (1992). Generalised eigenproblems arising in aggregated Markov processes allowing for time interval omission. Adv. Appl. Prob. 24, 302–321.CrossRefGoogle Scholar
Janssen, J., and Reinhard, J. M. (1982). Some duality results for a class of multivariate semi-Markov processes. J. Appl. Prob. 19, 90–98.Google Scholar
Keilson, J. (1969). On the matrix renewal function for Markov renewal processes. Ann. Math. Statist. 40, 1901–1907.Google Scholar
Keilson, J. (1979). Markov Chain Models–-Rarity and Exponentiality. Springer, New York.CrossRefGoogle Scholar
Keilson, J., and Wishart, D. G. M. (1964). A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.Google Scholar
Keilson, J., and Wishart, D. G. M. (1967). Addenda to processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 63, 187–193.Google Scholar
Kemeny, J., and Snell, J. (1976). Finite Markov Chains, 2nd edn. Springer, New York.Google Scholar
Kijima, S., and Kijima, H. (1987). Statistical analysis of channel current from a membrane patch. I. Some stochastic properties of ion channels or molecular systems in equilibrium. J. Theor. Biol. 128, 423–434.Google Scholar
Masuda, Y., and Sumita, U. (1987). Analysis of a counting process associated with a semi-Markov process: number of entries into a subset of state space. Adv. Appl. Prob. 19, 767–783.Google Scholar
Masuda, Y., and Sumita, U. (1991). A multivariate reward process defined on a semi-Markov process. J. Appl. Prob. 28, 360–373.Google Scholar
Pyke, R. (1961). Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.Google Scholar
Robert, C. P., Celeux, G., and Diebolt, J. (1993). Bayesian estimation of hidden Markov chains: a stochastic implementation. Statist. Prob. Lett. 16, 77–83.Google Scholar
Serfozo, R. F. (1975). Functional limit theorems for stochastic processes based on embedded processes. Adv. Appl. Prob. 7, 123–139.Google Scholar
Smith, A. F. M., and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. R. Statist. Soc. B 55, 3–23.Google Scholar
Stefanov, V. T., and Yeo, G. F. (1997). Explicit asymptotic results for open times in ion channel models. Adv. Appl. Prob. 29, 947–964.CrossRefGoogle Scholar