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The study develops a method to analyze the tails of sums of independent Banach-space valued random variables, suggesting new optimal inequalities.
We develop a new method to study the tails of a sum of independent mean zero Banach-space valued random variables (Xᵢ)i ≤ N. It relies on a new isoperimetric inequality for subsets of a product of probability spaces. In particular, we prove that for p ≥ 1, \|∑i ≤ N Xᵢ\|ₚ ≤ Kp/1 + log p(\|∑i ≤ N Xᵢ\|₁ + \|maxi ≤ N\|Xᵢ\|\|ₚ), where K is a universal constant. Other optimal inequalities for exponential moments are obtained.
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Michel Talagrand (1989) studied this question.
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