A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 120-134.

We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313-342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

DOI : 10.1051/ps/2011144
Classification : 60F05, 62F40, 62G07, 62M15
Mots clés : central limit theorem, Lindeberg method, weak dependence, bootstrap
@article{PS_2013__17__120_0,
     author = {Neumann, Michael H.},
     title = {A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics},
     journal = {ESAIM: Probability and Statistics},
     pages = {120--134},
     publisher = {EDP-Sciences},
     volume = {17},
     year = {2013},
     doi = {10.1051/ps/2011144},
     mrnumber = {3021312},
     zbl = {1291.60047},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2011144/}
}
TY  - JOUR
AU  - Neumann, Michael H.
TI  - A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics
JO  - ESAIM: Probability and Statistics
PY  - 2013
SP  - 120
EP  - 134
VL  - 17
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2011144/
DO  - 10.1051/ps/2011144
LA  - en
ID  - PS_2013__17__120_0
ER  - 
%0 Journal Article
%A Neumann, Michael H.
%T A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics
%J ESAIM: Probability and Statistics
%D 2013
%P 120-134
%V 17
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2011144/
%R 10.1051/ps/2011144
%G en
%F PS_2013__17__120_0
Neumann, Michael H. A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 120-134. doi : 10.1051/ps/2011144. http://www.numdam.org/articles/10.1051/ps/2011144/

[1] D.W.K. Andrews, Non-strong mixing autoregressive processes, J. Appl. Probab. 21 (1984) 930-934. | MR | Zbl

[2] J.M. Bardet, P. Doukhan, G. Lang and N. Ragache, Dependent Lindeberg central limit theorem and some applications. ESAIM : PS 12 (2008) 154-172. | Numdam | MR | Zbl

[3] P.J. Bickel and P. Bühlmann, A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5 (1999) 413-446. | MR | Zbl

[4] P. Billingsley, The Lindeberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12 (1961) 788-792. | MR | Zbl

[5] P. Billingsley, Convergence of Probability Measures. Wiley, New York (1968). | MR | Zbl

[6] C. Coulon-Prieur and P. Doukhan, A triangular central limit theorem under a new weak dependence condition. Stat. Probab. Lett. 47 (2000) 61-68. | MR | Zbl

[7] R. Dahlhaus, Fitting time series models to nonstationary processes. Ann. Stat. 25 (1997) 1-37. | MR | Zbl

[8] R. Dahlhaus, Local inference for locally stationary time series based on the empirical spectral measure. J. Econ. 151 (2009) 101-112. | MR

[9] J. Dedecker, A central limit theorem for stationary random fields. Probab. Theory Relat. Fields 110 (1998) 397-426. | MR | Zbl

[10] J. Dedecker and F. Merlevède, Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002) 1044-1081. | MR | Zbl

[11] J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes. Ann. Inst. Henri Poincaré Série B 36 (2000) 1-34. | Numdam | MR | Zbl

[12] J. Dedecker, P. Doukhan, G. Lang, J.R. León, S. Louhichi and C. Prieur, Weak Dependence : With Examples and Applications. Springer-Verlag. Lect. Notes Stat. 190 (2007). | MR | Zbl

[13] P. Doukhan, Mixing : Properties and Examples. Springer-Verlag. Lect. Notes Stat. 85 (1994). | MR | Zbl

[14] P. Doukhan and S. Louhichi, A new weak dependence condition and application to moment inequalities. Stoch. Proc. Appl. 84 (1999) 313-342. | MR | Zbl

[15] I.A. Ibragimov, Some limit theorems for stationary processes. Teor. Veroyatn. Primen. 7 (1962) 361-392 (in Russian). [English translation : Theory Probab. Appl. 7 (1962) 349-382]. | MR | Zbl

[16] I.A. Ibragimov, A central limit theorem for a class of dependent random variables. Teor. Veroyatnost. i Primenen. 8 (1963) 89-94 (in Russian). [English translation : Theor. Probab. Appl. 8 (1963) 83-89]. | MR | Zbl

[17] I.A. Ibragimov, A note on the central limit theorem for dependent random variables. Teor. Veroyatnost. i Primenen. 20 (1975) 134-140 (in Russian). [English translation : Theor. Probab. Appl. 20 (1975) 135-141]. | MR | Zbl

[18] J.W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung, Math. Zeitschr. 15 (1922) 211-225. | JFM | MR

[19] J.S. Liu, Siegel's formula via Stein's identities. Statist. Probab. Lett. 21 (1994) 247-251. | MR | Zbl

[20] H. Lütkepohl, Handbook of Matrices. Wiley, Chichester (1996). | MR | Zbl

[21] M.H. Neumann and E. Paparoditis, Goodness-of-fit tests for Markovian time series models : Central limit theory and bootstrap approximations. Bernoulli 14 (2008) 14-46. | MR | Zbl

[22] M.H. Neumann and E. Paparoditis, A test for stationarity. Manuscript (2011).

[23] M.H. Neumann and R. Von Sachs, Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 (1997) 38-76. | MR | Zbl

[24] E. Rio, About the Lindeberg method for strongly mixing sequences. ESAIM : PS 1 (1995) 35-61. | Numdam | MR | Zbl

[25] M. Rosenblatt, A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. | MR | Zbl

[26] M. Rosenblatt, Linear processes and bispectra. J. Appl. Probab. 17 (1980) 265-270. | MR | Zbl

[27] V.A. Volkonski and Y.A. Rozanov, Some limit theorems for random functions, Part I. Teor. Veroyatn. Primen. 4 (1959) 186-207 (in Russian). [English translation : Theory Probab. Appl. 4 (1959) 178-197]. | Zbl

Cité par Sources :