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Komlós-Major-Tusnády approximation for the general empirical process and Haar expansions of classes of functions. (English) Zbl 0810.60002
The author studies the rate of convergence in the central limit theorem for the general empirical processes indexed by a family of functions. He uses a modification of the coupling method proposed by J. Komlós, P. Major and G. Tusnády [in: Limit Theorems Probab. Theory, Keszthely 1974, Colloq. Math. Soc. János Bolyai 11, 149-165 (1975; Zbl 0342.60009)] for Gaussian approximation of the partial sum processes. The accuracy of the author’s approximation depends on some entropic characteristics of the family of functions and the accuracy of approximation of these functions by the Haar ones. As a corollary, the author obtains some well-known results as well as a new result for empirical characteristic functions.

MSC:
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
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