A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

Herold Dehling; Manfred Denker; Walter Philipp

doi:10.1007/BF00343899

A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

Herold Dehling, Manfred Denker, Walter Philipp

Mathematics

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20 Scopus citations

Abstract

A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (ℝ-valued) random variables {X_j, j≧1} and a kernel h: ℝ^m→H, m≧2, which is degenerate for the common distribution function of X_j, j≧1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=ℝ.

Original language	English (US)
Pages (from-to)	111-131
Number of pages	21
Journal	Probability Theory and Related Fields
Volume	72
Issue number	1
DOIs	https://doi.org/10.1007/BF00343899
State	Published - Apr 1986

All Science Journal Classification (ASJC) codes

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

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AB - A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (ℝ-valued) random variables {Xj, j≧1} and a kernel h: ℝm→H, m≧2, which is degenerate for the common distribution function of Xj, j≧1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=ℝ.

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A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

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