Extremal distributions for tail probabilities of sums

Two old conjectures from problem sections, one of which from SIAM Review, concern the question of finding distributions that maximize P(Sn ≤ t), where Sn is the sum of i.i.d. random variables X1, . . . , Xn on the interval [0, 1], satisfying E [X1] = m. In this paper a Lagrange multiplier technique is applied to this problem, yielding necessary conditions for distributions to be extremal, for arbitrary n. For n = 2, a complete solution is derived from them: extremal distributions are discrete and have one of the following supports, depending on m and t: {0, t}, {t− 1, 1}, {t/2, 1}, or {0, t, 1}. These results suffice to refute both conjectures. However, acquired insight naturally leads to a revised conjecture: that extremal distributions always have at most three support points and belong to a (for each n, specified) finite collection of two and three point distributions.