Summary
Let (x *i ) ni=1 denote the decreasing rearrangement of a sequence of real numbers (x i ) ni=1 . Then for everyi≠j, and every 1≦k≦n, the 2-nd order partial distributional derivatives satisfy the inequality,\(\frac{{\partial ^2 }}{{\partial x_i \partial x_j }}\left( {\sum\limits_{l = 1}^k {x_l^* } } \right) \leqq 0\). As a consequence we generalize the theorem due to Fernique and Sudakov. A generalization of Slepian's lemma is also a consequence of another differential inequality. We also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in ℝn proved by Gromov.
References
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Supported by NSF grant # DMS 8909745, and the USA-Israel Binational Science Foundation grant # 86-00074, and grant for the Promotion of Research at the Technion
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Gordon, Y. Majorization of Gaussian processes and geometric applications. Probab. Th. Rel. Fields 91, 251–267 (1992). https://doi.org/10.1007/BF01291425
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DOI: https://doi.org/10.1007/BF01291425
Keywords
- Real Number
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Gaussian Process