It is demonstrated that for each n ≧ 2 there exists a minimal universal constant, cn, such that, for any sequence of independent random variables {Xr, r ≧ 1} with finite variances, 𝔼[max1≦i≦nXi] – supT 𝔼XT ≦ cn√(n – 1) max1≦i≦n√(Var(Xi) where the supremum is over all stopping time T, 1 ≦ T ≦ n. Furthermore, cn ≦ 1/2 and lim infn→∞cn ≧ 0.439485….
