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Three theorems of multivariate empirical process

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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© 1976 Springer-Verlag

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Révész, P. (1976). Three theorems of multivariate empirical process. In: Gaenssler, P., Révész, P. (eds) Empirical Distributions and Processes. Lecture Notes in Mathematics, vol 566. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096882

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  • DOI: https://doi.org/10.1007/BFb0096882

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  • Print ISBN: 978-3-540-08061-9

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