Summary
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: ℝm→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.
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Dehling, H., Denker, M. & Philipp, W. A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics. Probab. Th. Rel. Fields 72, 111–131 (1986). https://doi.org/10.1007/BF00343899
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DOI: https://doi.org/10.1007/BF00343899
Keywords
- Distribution Function
- Hilbert Space
- Stochastic Process
- Probability Theory
- Statistical Theory