The Annals of Probability

A generalization of the Lindeberg principle

Sourav Chatterjee

Full-text: Open access

Abstract

We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.

Article information

Source
Ann. Probab. Volume 34, Number 6 (2006), 2061-2076.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1171377437

Digital Object Identifier
doi:10.1214/009117906000000575

Mathematical Reviews number (MathSciNet)
MR2294976

Zentralblatt MATH identifier
1117.60034

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G09: Exchangeability 15A52

Keywords
Lindeberg de Finetti Wigner exchangeable random matrices invariance principle semicircle law

Citation

Chatterjee, Sourav. A generalization of the Lindeberg principle. Ann. Probab. 34 (2006), no. 6, 2061--2076. doi:10.1214/009117906000000575. http://projecteuclid.org/euclid.aop/1171377437.


Export citation

References

  • Arnold, L. (1967). On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20 262--268.
  • Bai, Z. D. (1999). Methodologies in the spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611--677.
  • Baik, J. and Suidan, T. M. (2005). A GUE central limit theorem and universality of directed first and last passage site percolation. Int. Math. Res. Not. 6 325--337.
  • Bodineau, T. and Martin, J. (2005). A universality property for last-passage percolation paths close to the axis. Electron. Comm. Probab. 10 105--112.
  • Chatterjee, S. (2004). A simple invariance theorem. Available at http://arxiv.org/math.PR/0508213.
  • de Finetti, B. (1969). Sulla prosequibilità di processi aleatori scambiabili. Rend. Ist. Mat. Univ. Trieste 1 53--67.
  • Boutet de Monvel, A. and Khorunzhy, A. (1998). Limit theorems for random matrices. Markov Process. Related Fields 4 175--197.
  • Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences. Ann. Probab. 8 745--764.
  • Girko, V. L. (1988). Spectral Theory of Random Matrices. Nauka, Moscow.
  • Grenander, U. (1963). Probabilities on Algebraic Structures. Wiley, New York.
  • Khorunzhy, A. M., Khoruzhenko, B. A. and Pastur, L. A. (1996). Asymptotic properties of large random matrices with independent entries. J. Math. Phys. 37 5033--5060.
  • Lindeberg, J. W. (1922). Eine neue herleitung des exponentialgesetzes in der wahrscheinlichkeitsrechnung. Math. Z. 15 211--225.
  • Mossel, E., O'Donnell, R. and Oleszkiewicz, K. (2005). Noise stability of functions with low influences: Invariance and optimality. Available at http://arxiv.org/math.PR/0503503.
  • Pastur, L. A. (1972). The spectrum of random matrices. Teoret. Mat. Fiz. 10 102--112.
  • Paulauskas, V. and Račkauskas, A. (1989). Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces. Kluwer, Dordrecht.
  • Rotar, V. I. (1979). Limit theorems for polylinear forms. J. Multivariate Anal. 9 511--530.
  • Schenker, J. H. and Schulz-Baldes, H. (2005). Semicircle law and freeness for random matrices with symmetries or correlations. Math. Res. Lett. 12 531--542.
  • Suidan, T. (2006). A remark on a theorem of Chatterjee and last passage percolation. J. Phys. A 39 8977--8981.
  • Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models. Springer, Berlin.
  • Trotter, H. F. (1959). Elementary proof of the central limit theorem. Archiv der Mathem. 10 226--234.
  • Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325--327.
  • Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem. Clarendon Press, Oxford.
  • Zolotarev, V. M. (1977). Ideal metrics in the problem of approximating the distributions of sums of independent random variables. Theory Probab. Appl. 22 449--465.