Electronic Journal of Probability

Asymptotic variance of stationary reversible and normal Markov processes

George Deligiannidis, Magda Peligrad, and Sergey Utev

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We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in $n$

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 20, 26 pp.

Accepted: 3 March 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10 60J05 30C85

Markov chains asymptotic variance harmonic measure

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Deligiannidis, George; Peligrad, Magda; Utev, Sergey. Asymptotic variance of stationary reversible and normal Markov processes. Electron. J. Probab. 20 (2015), paper no. 20, 26 pp. doi:10.1214/EJP.v20-3183. https://projecteuclid.org/euclid.ejp/1465067126

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  • Billingsley, Patrick. Probability and measure. Third edition. Wiley Series in Probability and Mathematical Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1995. xiv+593 pp. ISBN: 0-471-00710-2.
  • Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9.
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1989. xx+494 pp. ISBN: 0-521-37943-1.
  • Borodin, A. N.; Ibragimov, I. A. Limit theorems for functionals of random walks. (Russian) Trudy Mat. Inst. Steklov. 195 (1994), Predel. Teoremy dlya Funktsional. ot Sluchain. Bluzh., 286 pp.; translation in Proc. Steklov Inst. Math. 1995, no. 2 (195), viii + 259 pp.
  • Bradley, Richard C. Introduction to strong mixing conditions. Vol. 3. Kendrick Press, Heber City, UT, 2007. xii+597 pp. ISBN: 0-9740427-8-1
  • Brockwell, Peter J.; Davis, Richard A. Time series: theory and methods. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 1991. xvi+577 pp. ISBN: 0-387-97429-6.
  • Cuny, Christophe; Lin, Michael. Pointwise ergodic theorems with rate and application to the CLT for Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 3, 710–733.
  • Deligiannidis, George; Utev, Sergey. Variance of partial sums of stationary sequences. Ann. Probab. 41 (2013), no. 5, 3606–3616.
  • Deligiannidis, G.; Utev, S. A. Computation of the asymptotics of the variance of the number of self-intersections of stable random walks using the Wiener-Darboux theory. (Russian) Sibirsk. Mat. Zh. 52 (2011), no. 4, 809–822; translation in Sib. Math. J. 52 (2011), no. 4, 639–650
  • Derriennic, Yves; Lin, Michael. The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Related Fields 119 (2001), no. 4, 508–528.
  • Doukhan, Paul; Massart, Pascal; Rio, Emmanuel. The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincaré Probab. Statist. 30 (1994), no. 1, 63–82.
  • Jewell, Nicholas P.; Bloomfield, Peter. Canonical correlations of past and future for time series: definitions and theory. Ann. Statist. 11 (1983), no. 3, 837–847.
  • Feller, William. An introduction to probability theory and its applications. Vol. I. 2nd ed. John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. xv+461 pp.
  • Giraudo, Davide; Volný, Dalibor. A strictly stationary $\beta$-mixing process satisfying the central limit theorem but not the weak invariance principle. Stochastic Process. Appl. 124 (2014), no. 11, 3769–3781.
  • Gordin, M. and Lifsic, B.: A remark about a Markov process with normal transition operator. phThird Vilnius Conf. Proba. Stat., Akad. Nauk Litovsk, Vilnius. 1, (1981), 147–148.
  • Geyer, C.J.: Practical Markov chain Monte Carlo. phStat. Sci. 7, (1992), 473–483.
  • Hardy, G. H.; Littlewood, J. E. Solution of the Cesàro summability problem for power-series and Fourier series. Math. Z. 19 (1924), no. 1, 67–96.
  • Häggström, Olle; Rosenthal, Jeffrey S. On variance conditions for Markov chain CLTs. Electron. Comm. Probab. 12 (2007), 454–464 (electronic).
  • Holzmann, Hajo. The central limit theorem for stationary Markov processes with normal generator-with applications to hypergroups. Stochastics 77 (2005), no. 4, 371–380.
  • Holzmann, Hajo. Martingale approximations for continuous-time and discrete-time stationary Markov processes. Stochastic Process. Appl. 115 (2005), no. 9, 1518–1529.
  • Ibragimov, I. A. Some limit theorems for stationary processes. (Russian) Teor. Verojatnost. i Primenen. 7 1962 361–392.
  • Kipnis, C.; Varadhan, S. R. S. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986), no. 1, 1–19.
  • Komorowski, Tomasz; Landim, Claudio; Olla, Stefano. Fluctuations in Markov processes. Time symmetry and martingale approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 345. Springer, Heidelberg, 2012. xviii+491 pp. ISBN: 978-3-642-29879-0.
  • Kontoyiannis, I.; Meyn, S. P. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 (2003), no. 1, 304–362.
  • Lawler, Gregory F. Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3.
  • Longla, Martial; Peligrad, Costel; Peligrad, Magda. On the functional central limit theorem for reversible Markov chains with nonlinear growth of the variance. J. Appl. Probab. 49 (2012), no. 4, 1091–1105.
  • Maxwell, Michael; Woodroofe, Michael. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000), no. 2, 713–724.
  • Merlevède, Florence; Peligrad, Magda. Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples. Ann. Probab. 41 (2013), no. 2, 914–960.
  • Olla, Stefano. Central limit theorems for tagged particles and for diffusions in random environment. Milieux aléatoires, 75–100, Panor. Synthèses, 12, Soc. Math. France, Paris, 2001.
  • Peligrad, Magda; Utev, Sergey. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005), no. 2, 798–815.
  • Rio, Emmanuel. Théorie asymptotique des processus aléatoires faiblement dépendants. (French) [Asymptotic theory of weakly dependent random processes] Mathématiques & Applications (Berlin) [Mathematics & Applications], 31. Springer-Verlag, Berlin, 2000. x+169 pp. ISBN: 3-540-65979-X.
  • Rio, Emmanuel. Moment inequalities for sums of dependent random variables under projective conditions. J. Theoret. Probab. 22 (2009), no. 1, 146–163.
  • Roberts, Gareth O.; Rosenthal, Jeffrey S. General state space Markov chains and MCMC algorithms. Probab. Surv. 1 (2004), 20–71.
  • Rudin, Walter. Functional analysis. Second edition. International Series in Pure and Applied Mathematics. McGraw-Hill, Inc., New York, 1991. xviii+424 pp. ISBN: 0-07-054236-8.
  • Seneta, Eugene. Regularly varying functions. Lecture Notes in Mathematics, Vol. 508. Springer-Verlag, Berlin-New York, 1976. v+112 pp.
  • Tierney, Luke. A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Probab. 8 (1998), no. 1, 1–9.
  • Tóth, Bálint. Persistent random walks in random environment. Probab. Theory Relat. Fields 71 (1986), no. 4, 615–625.
  • Tóth, Bálint. Comment on a theorem of M. Maxwell and M. Woodroofe. Electron. Commun. Probab. 18 (2013), no. 13, 4 pp.
  • Zhao, Ou; Woodroofe, Michael; Volný, Dalibor. A central limit theorem for reversible processes with nonlinear growth of variance. J. Appl. Probab. 47 (2010), no. 4, 1195–1202.