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Normal Approximation for Functions of Hidden Markov Models

Part of: Geometric probability and stochastic geometry Limit theorems Special processes

Published online by Cambridge University Press:  06 June 2022

Christian Houdré  and
George Kerchev
Christian Houdré*
Affiliation:
Georgia Institute of Technology
George Kerchev*
Affiliation:
Université du Luxembourg
*
*Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160, USA. Email address: houdre@math.gatech.edu
**Postal address: Université du Luxembourg, Unité de Recherche en Mathématiques, Maison du Nombre, 6 Avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duché du Luxembourg. Email address: gkerchev@gmail.com
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Abstract

The generalized perturbative approach is an all-purpose variant of Stein’s method used to obtain rates of normal approximation. Originally developed for functions of independent random variables, this method is here extended to functions of the realization of a hidden Markov model. In this dependent setting, rates of convergence are provided in some applications, leading, in each instance, to an extra log-factor vis-à-vis the rate in the independent case.

Keywords

Stein’s methodMarkov chainsgeneralized perturbative approachnormal approximationstochastic geometry

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60D05: Geometric probability and stochastic geometry
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 536 - 569
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Bousquet, O. and Houdré, C. (2019). Iterated jackknives and two-sided variance inequalities. In High Dimensional Probability VIII: The Oaxaca Volume (Progress in Probability 74), Birkhäuser, Cham, pp. 33–40.CrossRefGoogle Scholar
Chatterjee, S. (2008). A new method for normal approximation. Ann. Prob. 36, 1584–1610.CrossRefGoogle Scholar
Chatterjee, S. (2014). A short survey of Stein’s method. In Proceedings of the International Congress of Mathematicians: Seoul 2014, Vol. IV, Kyung Moon Sa, Seoul, pp. 1–24.Google Scholar
Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2014). Normal Approximation by Stein’s Method. Springer, Berlin, Heidelberg.Google Scholar
Chen, P., Shao, Q.-M. and Xu, L. (2020). A universal probability approximation method: Markov process approach. Preprint. Available at https://arxiv.org/abs/2011.10985.Google Scholar
Chu, D., Shao, Q.-M. and Zhang, Z. (2019). Berry–Esseen bounds for functionals of independent random variables. Presented at the Symposium in Memory of Charles Stein (1920–2016). Available at https://projecteuclid.org/journals/annals-of-probability/volume-47/issue-1/BerryEsseen-bounds-of-normal-and-nonnormal-approximation-for-unbounded-exchangeable/10.1214/18-AOP1255.full.Google Scholar
Durbin, R., Eddy, S., Krogh, A. and Mitchison, G. (1998). Biological Sequence Analysis. Cambridge University Press.CrossRefGoogle Scholar
Englund, G. (1981). A remainder term estimate for the normal approximation in classical occupancy. Ann. Prob. 9, 684–692.CrossRefGoogle Scholar
Gorodezky, I. and Pak, I. (2012). Generalized loop-erased random walks and approximate reachability. Random Structures Algorithms 44, 201–223.CrossRefGoogle Scholar
Grabchak, M., Kelbert, M. and Paris, Q. (2020). On the occupancy problem for a regime-switching model. J. Appl. Prob. 57, 53–77.CrossRefGoogle Scholar
Houdré, C. and Kerchev, G. (2019). On the rate of convergence for the length of the longest common subsequences in hidden Markov models. J. Appl. Prob. 56, 558–573.CrossRefGoogle Scholar
Houdré, C. and Ma, J. (2016). On the order of the central moments of the length of the longest common subsequences in random words. In High Dimensional Probability VII: The Cargèse Volume (Progress in Probability 71), Birkhäuser, Cham, pp. 105–136.CrossRefGoogle Scholar
Kendall, W. S. and Molchanov, I. (eds) (2010). New Perspectives in Stochastic Geometry. Oxford University Press.Google Scholar
Lachièze-Rey, R. and Peccati, G. (2017). New Berry–Esseen bounds for functionals of binomial point process. Ann. Appl. Prob. 27, 1992–2031.CrossRefGoogle Scholar
Paulin, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. Electron. J. Prob. 20, 32 pp.CrossRefGoogle Scholar
Rhee, W. and Talagrand, M. (1986). Martingale inequalities and the jackknife estimate of the variance. Statist. Prob. Lett. 4, 5–6.CrossRefGoogle Scholar
Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In STOC ’96: Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, pp. 296–303.CrossRefGoogle Scholar