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Hypercontraction Methods in Moment Inequalities for Series of Independent Random Variables in Normed Spaces

Stanislaw Kwapien and Jerzy Szulga
The Annals of Probability
Vol. 19, No. 1 (Jan., 1991), pp. 369-379
Stable URL: http://www.jstor.org/stable/2244266
Page Count: 11
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Hypercontraction Methods in Moment Inequalities for Series of Independent Random Variables in Normed Spaces
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Abstract

We prove that if (θk) is a sequence of i.i.d. real random variables then, for $1 < q < p$, the linear combinations of (θk) have comparable pth and qth moments if and only if the joint distribution of (θk) is (p, q)-hypercontractive. We elaborate hypercontraction methods in a new proof of the inequality $\bigg(E\bigg\|\sum_i X_i\bigg\|^p\bigg)^{1/p} \leq C_p\bigg(E\big\|\sum_i X_i\bigg\| + \big(E\sup_i\|X_i\|^p\big)^{1/p}\bigg),$ where (Xi) is a sequence of independent zero-mean random variables with values in a normed space, and Cp ≈ p/ln p.

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