Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion
Correlation Inequalities and Applications to Vector-Valued Gaussian Random Variables and Fractional Brownian Motion
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相关不等式及其在向量值高斯随机变量和分数布朗运动中的应用
DOI：
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发表时间：
2009
期刊：
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通讯作者：
M. Veraar
中科院分区：
文献类型：
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作者：
M. Veraar
本文推广了Kolmogorov和Rozanov关于向量值高斯随机变量的相关不等式。将这些不等式应用于高斯随机变量序列和高斯过程。对于满足相关假设的高斯随机变量序列，我们证明了Borel-坎特利引理、极大不等式和几个大数定律。这推广了Beśka和Ciesielski以及Hytönen和作者的结果。在本文的第二部分中，我们考虑了一类在p阶矩上α-Hölder连续的向量值高斯过程。对于这些过程，我们得到了α阶路径的Besov正则性。我们还得到了Besov范数中矩的估计。特别地，这些结果被应用于向量值分数布朗运动。这些结果推广了Ciesielski、Kerkyacharian和Roynette以及Hytönen和作者的早期工作。
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.