Central Limit Theorems

  • Tomasz Komorowski
  • Claudio Landim
  • Stefano Olla
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 345)


We extend the ideas introduced in Chap.  1 to continuous time Markov processes on general state spaces with a stationary ergodic measure allowed to be non-reversible. We first prove a central limit theorem for continuous time ergodic martingales. Then we prove the central limit theorem for functions V such that certain convergence properties of the corresponding resolvent (λL)−1 V are satisfied, where L is the generator of the process. We move from reversible processes, where, like in Chap.  1, finiteness of the limit variance imply these convergence conditions, to processes satisfying a sector condition or more generally a graded sector condition. This last condition will be motivated later in Parts II and III by important examples (exclusion processes and diffusions in Gaussian random fields). Normal generators are other examples, that turn out to be important later on for the diffusions in time dependent environments.


Markov Process Central Limit Theorem Sector Condition Invariance Principle Additive Functional 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Blumenthal RM, Getoor RK (1968) Markov processes and potential theory. Pure and applied mathematics, vol 29. Academic Press, New York zbMATHGoogle Scholar
  2. Bolthausen E (1980) The Berry-Esseen theorem for functionals of discrete Markov chains. Z Wahrscheinlichkeitstheor Verw Geb 54(1):59–73 MathSciNetzbMATHCrossRefGoogle Scholar
  3. Bolthausen E (1982) The Berry-Esseén theorem for strongly mixing Harris recurrent Markov chains. Z Wahrscheinlichkeitstheor Verw Geb 60(3):283–289 MathSciNetzbMATHCrossRefGoogle Scholar
  4. Chatterjee S, Diaconis P, Meckes E (2005) Exchangeable pairs and Poisson approximation. Probab Surv 2:64–106 (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  5. Cheeger J (1970) A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in analysis (Papers dedicated to Salomon Bochner, 1969). Princeton University Press, Princeton, pp 195–199 Google Scholar
  6. Chung KL, Walsh JB (2005) Markov processes, Brownian motion, and time symmetry. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 249, 2nd edn. Springer, New York zbMATHGoogle Scholar
  7. De Masi A, Ferrari PA, Goldstein S, Wick WD (1989) An invariance principle for reversible Markov processes. Applications to random motions in random environments. J Stat Phys 55(3–4):787–855 zbMATHCrossRefGoogle Scholar
  8. Derriennic Y, Lin M (1996) Sur le théorème limite central de Kipnis et Varadhan pour les chaînes réversibles ou normales. C R Acad Sci Paris Sér I Math 323(9):1053–1057 MathSciNetzbMATHGoogle Scholar
  9. Derriennic Y, Lin M (2001a) The central limit theorem for Markov chains with normal transition operators, started at a point. Probab Theory Relat Fields 119(4):508–528 MathSciNetzbMATHCrossRefGoogle Scholar
  10. Diaconis P, Holmes S (eds) (2004) Stein’s method: expository lectures and applications. IMS lect notes monogr ser, vol 46. Inst Math Stat, Beachwood. Papers from the workshop on Stein’s method held at Stanford University, Stanford, CA, 1998 zbMATHGoogle Scholar
  11. Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics: probability and mathematical statistics. Wiley, New York zbMATHCrossRefGoogle Scholar
  12. Goldstein S (1995) Antisymmetric functionals of reversible Markov processes. Ann Inst Henri Poincaré Probab Stat 31(1):177–190 zbMATHGoogle Scholar
  13. Helland IS (1982) Central limit theorems for martingales with discrete or continuous time. Scand J Stat 9(2):79–94 MathSciNetzbMATHGoogle Scholar
  14. Horváth I, Tóth B, Vető B (2010) Diffusive limits for “true” (or myopic) self-avoiding random walks and self-repellent Brownian polymers in d≥3. arXiv:1009.0401
  15. Jacod J, Shiryaev AN (1987) Limit theorems for stochastic processes. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 288. Springer, Berlin zbMATHCrossRefGoogle Scholar
  16. Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Graduate texts in mathematics, vol 113, 2nd edn. Springer, New York zbMATHGoogle Scholar
  17. Kipnis C (1986) Central limit theorems for infinite series of queues and applications to simple exclusion. Ann Probab 14(2):397–408 MathSciNetzbMATHCrossRefGoogle Scholar
  18. Kipnis C, Landim C (1999) Scaling limits of interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 320. Springer, Berlin zbMATHCrossRefGoogle Scholar
  19. Kipnis C, Varadhan SRS (1986) Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun Math Phys 104(1):1–19 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Komorowski T, Olla S (2003b) On the sector condition and homogenization of diffusions with a Gaussian drift. J Funct Anal 197(1):179–211 MathSciNetzbMATHCrossRefGoogle Scholar
  21. Kontoyiannis I, Meyn SP (2003) Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann Appl Probab 13(1):304–362 MathSciNetzbMATHCrossRefGoogle Scholar
  22. Krengel U (1985) Ergodic theorems: with a supplement by Antoine Brunel. de Gruyter studies in mathematics, vol 6. de Gruyter, Berlin zbMATHCrossRefGoogle Scholar
  23. Landim C, Yau HT (1997) Fluctuation-dissipation equation of asymmetric simple exclusion processes. Probab Theory Relat Fields 108(3):321–356 MathSciNetzbMATHCrossRefGoogle Scholar
  24. Lawler GF, Sokal AD (1988) Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality. Trans Am Math Soc 309(2):557–580 MathSciNetzbMATHGoogle Scholar
  25. Levin DA, Peres Y, Wilmer EL (2009) Markov chains and mixing times: with a chapter by James G Propp and David B Wilson. Am Math Soc, Providence Google Scholar
  26. Lezaud P (2001) Chernoff and Berry-Esséen inequalities for Markov processes. ESAIM Probab Stat 5:183–201 (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  27. Liggett TM (1985) Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 276. Springer, New York zbMATHCrossRefGoogle Scholar
  28. Olla S (1994a) Homogenization of diffusion processes in random fields. Centre de mathématiques appliquées, ecole polytechnique. Available at
  29. Osada H (1998a) An invariance principle for Markov processes and Brownian particles with singular interaction. Ann Inst Henri Poincaré Probab Stat 34(2):217–248 MathSciNetzbMATHCrossRefGoogle Scholar
  30. Osada H, Saitoh T (1995) An invariance principle for non-symmetric Markov processes and reflecting diffusions in random domains. Probab Theory Relat Fields 101(1):45–63 MathSciNetzbMATHCrossRefGoogle Scholar
  31. Rebolledo R (1980) Central limit theorems for local martingales. Z Wahrscheinlichkeitstheor Verw Geb 51(3):269–286 MathSciNetzbMATHCrossRefGoogle Scholar
  32. Rogers LCG, Williams D (2000) Diffusions, Markov processes, and martingales, vol 1: foundations. Cambridge mathematical library. Cambridge University Press, Cambridge. Reprint of the second 1994 edition zbMATHGoogle Scholar
  33. Sethuraman S, Varadhan SRS, Yau HT (2000) Diffusive limit of a tagged particle in asymmetric simple exclusion processes. Commun Pure Appl Math 53(8):972–1006 MathSciNetzbMATHCrossRefGoogle Scholar
  34. Tóth B (1986) Persistent random walks in random environment. Probab Theory Relat Fields 71(4):615–625 MathSciNetzbMATHCrossRefGoogle Scholar
  35. Varadhan SRS (1995) Self-diffusion of a tagged particle in equilibrium for asymmetric mean zero random walk with simple exclusion. Ann Inst Henri Poincaré Probab Stat 31(1):273–285 MathSciNetzbMATHGoogle Scholar
  36. Whitt W (2007) Proofs of the martingale FCLT. Probab Surv 4:268–302 MathSciNetzbMATHCrossRefGoogle Scholar
  37. Wu L (1999) Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes. Ann Inst Henri Poincaré Probab Stat 35(2):121–141 zbMATHCrossRefGoogle Scholar
  38. Yosida K (1995) Functional analysis. Classics in mathematics. Springer, Berlin. Reprint of the sixth 1980 edition Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tomasz Komorowski
    • 1
    • 2
  • Claudio Landim
    • 3
    • 4
  • Stefano Olla
    • 5
  1. 1.Institute of MathematicsMaria Curie-Skłodowska UniversityLublinPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Instituto de Matemática (IMPA)Rio de JaneiroBrazil
  4. 4.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance
  5. 5.CEREMADE, CNRS UMR 7534Université Paris-DauphineParisFrance

Personalised recommendations