Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions

• Autorzy: Albert Baernstein II , Robert C. Culverhouse
• Języki publikacji: EN

Abstrakty

EN

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 ℝⁿ, and $a_{i},b_{i}$ are real constants. We prove that if ${b²_{i}}$ is majorized by ${a²_{i}}$ in the sense of Hardy-Littlewood-Pólya, and if Φ: ℝⁿ → ℝ is continuous and bisubharmonic, then EΦ(X) ≤ EΦ(Y). Consequences include most of the known sharp $L²-L^{p}$ 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 ℝⁿ instead of on the unit sphere.

Kategorie tematyczne

• 60G50: Sums of independent random variables; random walks
• 60E15: Inequalities; stochastic orderings

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2002
• Tom: 152
• Numer: 3
• Strony: 231-248

Daty

wydano

2002

Twórcy

autor

Albert Baernstein II

• Mathematics Department, Washington University, St. Louis, MO 63130, U.S.A.

autor

Robert C. Culverhouse

• Department of Psychiatry, Washington University Medical School, St. Louis, MO 63110, U.S.A.

Identyfikatory

DOI

10.4064/sm152-3-3