What question did this study set out to answer?

This research aims to explore the tails of the sum of independent mean zero Banach-space valued random variables.

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October 1, 1989Open Access

Isoperimetry and Integrability of the Sum of Independent Banach-Space Valued Random Variables

Key Points

This research aims to explore the tails of the sum of independent mean zero Banach-space valued random variables.
Developed a new isoperimetric inequality for subsets of product probability spaces.
Proved a bound on the p-norm of the sum in terms of the 1-norm and maximum norms.
Derived optimal inequalities for exponential moments.
Established the inequality \( \bigg\|\sum_{i \leq N} X_i\bigg\|_p \leq \frac{Kp}{1 + \log p} \bigg(\bigg\|\sum_{i \leq N} X_i\bigg\|_1 + \|\max_{i \leq N}\|X_i\|\|_p\bigg)\; (K: \text{universal constant}).
Identified optimal inequalities for exponential moments.

Abstract

We develop a new method to study the tails of a sum of independent mean zero Banach-space valued random variables (Xᵢ) ₈ ₍. It relies on a new isoperimetric inequality for subsets of a product of probability spaces. In particular, we prove that for p 1, \|₈ ₍ Xᵢ\|ₚ Kp1 + p (\|₈ ₍ Xᵢ\|₁ + \|₈ ₍\|Xᵢ\|\|ₚ), where K is a universal constant. Other optimal inequalities for exponential moments are obtained.

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