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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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