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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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