Abstract
Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞.
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