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Three theorems of multivariate empirical process

  • P. Révész
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 566)

Abstract

Let x 1 , x 2 ,... be a sequence of independent r.v.'s uniformly distributed over the unit cube I d of the d-dimensional Euclidean Space. Further let F n be the empirical distribution function based on the sample x 1 ,x 2 ,...,x n and let
$$\alpha _n (x) = n^{1/2} (F_n (x) - x_1 x_2 ...x_d )(x = (x_1 ,x_2 ,...,x_d ) \in _I ^d )$$
be the empirical process. The properties of the stochastic set function \(\alpha _n (A) = \int\limits_A {d\alpha _n }\) are investigated when A runs over a class of Borel sets of I d . Let A be the set of Borel sets of I d having d-times differentiable boundaries. Then a large deviation theorem and a law of iterated logarithm are proved for \(\mathop {\sup }\limits_{A \in A} \alpha _n (A)\). A strong invariance principle (uniform over A) is also formulated.

Keywords

Invariance Principle Empirical Measure Empirical Process Iterate Logarithm Empirical Distribution Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1976

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  • P. Révész

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