Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T18:39:33.887Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem for triangular arrays of weakly dependent random variables, with applications in statistics

Published online by Cambridge University Press:  08 February 2013

Michael H. Neumann
Michael H. Neumann*
Affiliation:
Friedrich-Schiller-Universität Jena, Institut für Stochastik, Ernst-Abbe-Platz 2, 07743 Jena, Germany. michael.neumann@uni-jena.de
Get access

Abstract

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.

Keywords

Central limit theoremLindeberg methodweak dependencebootstrap.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 120 - 134
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, D.W.K., Non-strong mixing autoregressive processes, J. Appl. Probab. 21 (1984) 930–934. Google Scholar
Bardet, J.M., Doukhan, P., Lang, G. and Ragache, N., Dependent Lindeberg central limit theorem and some applications. ESAIM : PS 12 (2008) 154–172. Google Scholar
Bickel, P.J. and Bühlmann, P., A new mixing notion and functional central limit theorems for a sieve bootstrap in time series. Bernoulli 5 (1999) 413–446. Google Scholar
Billingsley, P., The Lindeberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12 (1961) 788–792. Google Scholar
P. Billingsley, Convergence of Probability Measures. Wiley, New York (1968).
Coulon-Prieur, C. and Doukhan, P., A triangular central limit theorem under a new weak dependence condition. Stat. Probab. Lett. 47 (2000) 61–68. Google Scholar
Dahlhaus, R., Fitting time series models to nonstationary processes. Ann. Stat. 25 (1997) 1–37. Google Scholar
Dahlhaus, R., Local inference for locally stationary time series based on the empirical spectral measure. J. Econ. 151 (2009) 101–112. Google Scholar
Dedecker, J., A central limit theorem for stationary random fields. Probab. Theory Relat. Fields 110 (1998) 397–426. Google Scholar
Dedecker, J. and Merlevède, F., Necessary and sufficient conditions for the conditional central limit theorem. Ann. Probab. 30 (2002) 1044–1081. Google Scholar
Dedecker, J. and Rio, E., On the functional central limit theorem for stationary processes. Ann. Inst. Henri Poincaré Série B 36 (2000) 1–34. Google Scholar
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).
P. Doukhan, Mixing : Properties and Examples. Springer-Verlag. Lect. Notes Stat. 85 (1994).
Doukhan, P. and Louhichi, S., A new weak dependence condition and application to moment inequalities. Stoch. Proc. Appl. 84 (1999) 313–342. Google Scholar
Ibragimov, I.A., Some limit theorems for stationary processes. Teor. Veroyatn. Primen. 7 (1962) 361–392 (in Russian). [English translation : Theory Probab. Appl. 7 (1962) 349–382]. Google Scholar
Ibragimov, I.A., 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]. Google Scholar
Ibragimov, I.A., 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]. Google Scholar
Lindeberg, J.W., Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung, Math. Zeitschr. 15 (1922) 211–225. Google Scholar
Liu, J.S., Siegel’s formula via Stein’s identities. Statist. Probab. Lett. 21 (1994) 247–251. Google Scholar
H. Lütkepohl, Handbook of Matrices. Wiley, Chichester (1996).
Neumann, M.H. and Paparoditis, E., Goodness-of-fit tests for Markovian time series models : Central limit theory and bootstrap approximations. Bernoulli 14 (2008) 14–46. Google Scholar
M.H. Neumann and E. Paparoditis, A test for stationarity. Manuscript (2011).
Neumann, M.H. and von Sachs, R., Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra. Ann. Statist. 25 (1997) 38–76. Google Scholar
Rio, E., About the Lindeberg method for strongly mixing sequences. ESAIM : PS 1 (1995) 35–61. Google Scholar
Rosenblatt, M., A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43–47. Google Scholar
Rosenblatt, M., Linear processes and bispectra. J. Appl. Probab. 17 (1980) 265–270. Google Scholar
Volkonski, V.A. and Rozanov, Y.A., 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]. Google Scholar