### Abstract

Let X = ∑_{i=1}^{k} a_{i}U_{i}, Y = ∑_{i=1}^{k} b_{i}U_{i}, where the U_{i} are independent random vectors, each uniformly distributed on the unit sphere in ℝ^{n}, and a_{i}, b_{i} are real constants. We prove that if {b_{i}^{2}} is majorized by {a_{i}^{2} } in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝ^{n} → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp L^{2}-L^{p} Khinchin inequalities for sums of the form X. For radial Φ, bisubharmonicity is necessary as well as sufficient for the majorization inequality to always hold. Counterparts to the majorization inequality exist when the U_{i} are uniformly distributed on the unit ball of ℝ^{n} instead of on the unit sphere.

Original language | English |
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Pages (from-to) | 231-248 |

Number of pages | 18 |

Journal | Studia Mathematica |

Volume | 152 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2002 |

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## Cite this

*Studia Mathematica*,

*152*(3), 231-248. https://doi.org/10.4064/sm152-3-3