On the weak invariance principle and its quenched version under projective criteria | Vidéo | Carmin.tv

00:00:00 / 00:00:00

Chapters

On the weak invariance principle and its quenched version under projective criteria

By Florence Merlevède

Appears in collection : Limit theorems in dynamics and applications / Théorèmes limites en dynamique et applications

In this talk, we shall first review some projective criteria under which the central limit theorem holds. The projective criteria considered will be the Heyde criterion, the Hannan criterion, the Maxwell-Woodroofe condition and the Dedecker-Rio's condition. We shall also investigate under which projective criteria the reinforced versions of the CLT such as the weak invariance principle or the quenched CLT (and its functional form) still hold.

Information about the video

Date of recording 08/07/2014
Date of publication 22/07/2014
Institution CIRM
Licence CC BY NC ND
Language English
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.18538103
Cite this video Merlevède, Florence (08/07/2014). On the weak invariance principle and its quenched version under projective criteria. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.18538103
URL https://dx.doi.org/10.24350/CIRM.V.18538103

Domain(s)

Probability

Bibliography

Borodin, A.N. and Ibragimov, I.A. Limit theorems for functionals of random walks. Trudy Mat. Inst. Steklov. 195. (1994). Transl. into English: Proc. Steklov Inst. Math. (1995) - https://www.zbmath.org/?q=an:0855.60001
Bradley, R. C. On quantiles and the central limit question for strongly mixing sequences. Dedicated to Murray Rosenblatt. J. Theoret. Probab. 10 (1997), 507-555 - https://www.zbmath.org/?q=an:0887.60028
Chen, X. Limit theorems for functionals of ergodic Markov chains with general state space. Memoirs of the AMS, 139 nr 664 (1999) - http://dx.doi.org/10.1090/memo/0664
Cuny, C. and Merlevede, F. On martingale approximations and the quenched weak invariance principle. Ann. Probab. 42 (2014), 760-793 - http://dx.doi.org/10.1214/13-aop856
Cuny, C. and Peligrad, M. Central limit theorem started at a point for stationary processes and additive functionals of reversible Markov chains. J. Theor. Probability. 25 (2012), 171-188 - http://dx.doi.org/10.1007/s10959-010-0321-8
Cuny, C. and Volny, D. A quenched invariance principle for stationary processes. arXiv:1202.4875, accepted for publication in ALEA Lat. Am. J. Probab. Math. Stat. (2012) - https://www.zbmath.org/?q=an:1277.60056
Dedecker, J., Merlevede, F. and Peligrad, M. A quenched weak invariance principle. Ann. Inst. H. Poincare Probab. Statist. 50 (2014), 872-898 - http://dx.doi.org/10.1214/13-aihp553
Dedecker, J., Merlevede, F. and Volny, D. On the weak invariance principle for non- adapted sequences under projective criteria. J. Theoret. Probab. 20 (2007), 971-1004 - http://dx.doi.org/10.1007/s10959-007-0090-1
Dedecker, J. and Rio, E. On the functional central limit theorem for stationary processes, Ann. Inst. H. Poincare Probab. Statist. 36 (2000), 1-34 - https://www.zbmath.org/?q=an:0949.60049
Derriennic, Y. and Lin, M. The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Relat. Fields. 119 (2001), 508- 528 - http://dx.doi.org/10.1007/pl00008769
Derriennic, Y. and Lin, M. The central limit theorem for Markov chains started at a point, Probab. Theory Relat. Fields. 125 (2003), 73-76 - http://dx.doi.org/10.1007/s004400200215
Doukhan, P., Massart, P. and Rio, E. The functional central limit theorem for strongly mixing processes. Ann. Inst. H. Poincare Probab. Statist. 30 (1994), 63-82 - https://www.zbmath.org/?q=an:0790.60037
Durieu, O. Independence of four projective criteria for the weak invariance principle. ALEA Lat. Am. J. Probab. Math. Stat. 5 (2009), 21-26 - https://www.zbmath.org/?q=an:1169.60005
Durieu, O. and Volny, D. Comparison between criteria leading to the weak invariance principle. Ann. Inst. Henri Poincare Probab. Stat. 44 (2008), 324-340 - http://dx.doi.org/10.1214/07-aihp123
Gordin, M. I. and Lifsic, B. A. The central limit theorem for stationary Markov processes, Soviet Math. Dokl. 19 (1978), 392-394 - https://www.zbmath.org/?q=an:0395.60057
Gordin, M. and Peligrad, M. On the functional central limit theorem via martingale approximation. Bernoulli 17 (2011), 424-440 - http://dx.doi.org/10.3150/10-bej276
Hannan, E. J. The central limit theorem for time series regression. Stoch. Processes Appl. 9 (1979), 281-289 - http://dx.doi.org/10.1016/0304-4149(79)90050-4
Heyde, C. C. On the central limit theorem for stationary processes. Z. Wahrsch. verw. Gebiete. 30 (1974), 315-320 - http://dx.doi.org/10.1007/bf00532619
Heyde, C. C. On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. 12 (1975), 1-8 - http://dx.doi.org/10.1017/s0004972700023583
Kipnis, C., Varadhan, S.R.S.. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104 (1986), 1-19 - http://dx.doi.org/10.1007/bf01210789
Maxwell, M. and Woodroofe, M. Central limit theorem for additive fonctionals of Markov chains. Ann. Probab. 28 (2000), 713-724 - http://dx.doi.org/10.1214/aop/1019160258
Volny, D. and Woodroofe, M. An example of non-quenched convergence in the conditional central limit theorem for partial sums of a linear process. Dependence in analysis, probability and number theory (The Phillipp memorial volume), Kendrick Press. 317-323, (2010) - https://www.zbmath.org/?q=an:1213.60051
Wu, W.-B. and Woodroofe, M. Martingale approximations for sums of stationary processes. Ann. Probab. 32 (2004), 1674-1690 - http://dx.doi.org/10.1214/009117904000000351
Zhao, O. and Woodroofe, M. On martingale approximations. Ann. Appl. Probab. 18 (2008), 1831-1847 - http://dx.doi.org/10.1214/07-aap505

MSC codes

Last related questions on MathOverflow

You have to connect your Carmin.tv account with mathoverflow to add question

Ask a question on MathOverflow

Copyright Carmin.tv 2026

Give feedback