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A generalization of the Lindeberg principle. (English) Zbl 1117.60034
The author establishes the following inequality: Let $${\mathbf X}= (X_1,\dots, X_n)$$ and $${\mathbf Y}= (Y_1,\dots, Y_n)$$ be random vectors in $$\mathbb{R}^n$$ with the $$Y_i$$ being independent. Let $$f: \mathbb{R}^n\to\mathbb{R}$$ be a thrice continuously differentiable function whose $$r$$-fold derivatives are uniformly bounded by $$L_r(f)$$ $$(r= 1,2,3)$$. For each $$1\leq i\leq n$$, let $A_i= E|E(X_i|X_1,\dots, X_{i-1})- EY_i|,\quad B_i= E|E(X^2_i|X_1,\dots, X_{i-1})- EY^2_i|,$
$\max_{1\leq i\leq n}|E|X_i|^3+ E|Y_i|^3| \leq M_3.$ Then $|Ef({\mathbf X})- Ef({\mathbf Y})|\leq \sum^n_{i=1} (A_i L_1(f)+ \textstyle{{1\over 2}} B_i L_2(f))+ \textstyle{{1\over 6}} nL_3(f) M_3.$ Using this result, the author shows a similar one for smooth functions of exchangeable random variables, which enables us to get the limiting spectral distributions of Wigner matrices with exchangeable entries.

##### MSC:
 60F17 Functional limit theorems; invariance principles 60G09 Exchangeability for stochastic processes 15B52 Random matrices (algebraic aspects)
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##### References:
 [1] Arnold, L. (1967). On the asymptotic distribution of the eigenvalues of random matrices. J. Math. Anal. Appl. 20 262–268. · Zbl 0246.60029 [2] Bai, Z. D. (1999). Methodologies in the spectral analysis of large-dimensional random matrices, a review. Statist. Sinica 9 611–677. · Zbl 0949.60077 [3] 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. · Zbl 1136.60313 [4] Bodineau, T. and Martin, J. (2005). A universality property for last-passage percolation paths close to the axis. Electron. Comm. Probab. 10 105–112. · Zbl 1111.60068 [5] Chatterjee, S. (2004). A simple invariance theorem. Available at http://arxiv.org/math.PR/0508213. [6] de Finetti, B. (1969). Sulla prosequibilità di processi aleatori scambiabili. Rend. Ist. Mat. Univ. Trieste 1 53–67. · Zbl 0218.60106 [7] Boutet de Monvel, A. and Khorunzhy, A. (1998). Limit theorems for random matrices. Markov Process. Related Fields 4 175–197. · Zbl 0917.60040 [8] Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences. Ann. Probab. 8 745–764. · Zbl 0434.60034 [9] Girko, V. L. (1988). Spectral Theory of Random Matrices . Nauka, Moscow. · Zbl 0656.15012 [10] Grenander, U. (1963). Probabilities on Algebraic Structures . Wiley, New York. · Zbl 0131.34804 [11] 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. · Zbl 0866.15014 [12] Lindeberg, J. W. (1922). Eine neue herleitung des exponentialgesetzes in der wahrscheinlichkeitsrechnung. Math. Z. 15 211–225. · JFM 48.0602.04 [13] 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. · Zbl 1201.60031 [14] Pastur, L. A. (1972). The spectrum of random matrices. Teoret. Mat. Fiz. 10 102–112. [15] Paulauskas, V. and Račkauskas, A. (1989). Approximation Theory in the Central Limit Theorem. Exact Results in Banach Spaces . Kluwer, Dordrecht. · Zbl 0715.60023 [16] Rotar, V. I. (1979). Limit theorems for polylinear forms. J. Multivariate Anal. 9 511–530. · Zbl 0426.62013 [17] 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. · Zbl 1095.82004 [18] Suidan, T. (2006). A remark on a theorem of Chatterjee and last passage percolation. J. Phys. A 39 8977–8981. · Zbl 1148.82014 [19] Talagrand, M. (2003). Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Models . Springer, Berlin. · Zbl 1033.82002 [20] Trotter, H. F. (1959). Elementary proof of the central limit theorem. Archiv der Mathem. 10 226–234. · Zbl 0086.34002 [21] Wigner, E. P. (1958). On the distribution of the roots of certain symmetric matrices. Ann. of Math. (2) 67 325–327. JSTOR: · Zbl 0085.13203 [22] Wilkinson, J. H. (1965). The Algebraic Eigenvalue Problem . Clarendon Press, Oxford. · Zbl 0258.65037 [23] 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. · Zbl 0385.60025
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