An improvement of Hoffmann-Jorgensen's inequality

Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x is an element of B set \\X\\ = sup(Lambda is an element ofF) Lambda>(*) over bar * (x) (\\.\\ is at least a seminorm on B) , We give probability estimates for the tail probability of S-n* = max(1 le ss than or equal tok less than or equal ton) \\Sigma (k)(j=1) X-j\\ where { X-i}(i=1)(n) are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S-n* exceeds a t hreshold defined in terms of Sigma (k)(j=1) Y-(j), where Y-(r) denotes the rth largest term of {\\X-i\\}(i=1)(n). Using these tail estimates, essentia lly all the known results concerning the order of magnitude or finiteness o f quantities such as E Phi>(*) over bar *(\\S-n\\) and E Phi>(*) over bar * (S-n*) follow (for any fixed 1 less than or equal to n less than or equal to infinity). Included in this paper are uniform L-P bounds of S-n* which a re within a factor of 4 for all p greater than or equal to 1 and within a f actor of 2 in the limit as p --> infinity.