Abstract
In this paper we extend certain correlation inequalities for vector-valued Gaussian random variables due to Kolmogorov and Rozanov. The inequalities are applied to sequences of Gaussian random variables and Gaussian processes. For sequences of Gaussian random variables satisfying a correlation assumption, we prove a Borel-Cantelli lemma, maximal inequalities and several laws of large numbers. This extends results of Beśka and Ciesielski and of Hytönen and the author. In the second part of the paper we consider a certain class of vector-valued Gaussian processes which are α-Hölder continuous in p-th moment. For these processes we obtain Besov regularity of the paths of order α. We also obtain estimates for the moments in the Besov norm. In particular, the results are applied to vector-valued fractional Brownian motions. These results extend earlier work of Ciesielski, Kerkyacharian and Roynette and of Hytönen and the author.
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This work originated during the author’s stay at the IMPAN institute in Warsaw which was supported by the Research Training Network MRTN-CT-2004-511953. Part of the work has been carried out during the author’s stay at the University of Karlsruhe as an Alexander von Humboldt fellow.
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Veraar, M. Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion. Potential Anal 30, 341–370 (2009). https://doi.org/10.1007/s11118-009-9118-8
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DOI: https://doi.org/10.1007/s11118-009-9118-8
Keywords
- Correlation inequalities
- Gebelein’s inequality
- Gaussian random variables
- Maximal inequalities
- Law of large numbers
- Type and cotype
- Gaussian processes
- Fractional Brownian motion
- Besov–Orlicz spaces
- Sample path
- Non-separable Banach space