标题： 独立Rademacher随机变量和的均值偏差的精确上界 显示英文标题

标题： Sharp upper bounds for the deviations from the mean of the sum of independent Rademacher random variables

摘要： 对于球面 S^{n-1}上的一个固定单位向量 a=(a_1,a_2,...,a_n)，即 sum_{i=1}^n a_i^2=1，我们考虑 2^n 个符号向量 epsilon=(epsilon_1,epsilon_2,...,epsilon_n) 在 {-1,1}^n 中以及相应的标量积 a.epsilon=sum_{i=1}^n a_i epsilon_i。 Holtzman 和 Kleitman 提出了以下猜想。 该猜想指出，在所有形式为 sum +/- a_i 的 2^n 个和中，其绝对值 |sum_{i=1}^n +/- a_i|>1 的数量不超过 |sum_{i=1}^n +/- a_i| <= 1 的数量。 该结果本身就有意义，而且在概率论和几何学中也有吸引人的重新表述。 在本文中，我们将解决这个问题在所有 a 都相等的均匀情况下的扩展。 更准确地说，对于 S_n 是 n 个独立的 Rademacher 随机变量的和，我们将给出几个 xi 值的概率 P_n:=P{负xi乘以平方根{n}小于等于S_n小于等于xi乘以平方根{n}}的精确下界。 这与参数为 n 和 p=1/2 的二项分布有明显的联系。 得到的下界是紧的，并且比例如应用切比雪夫不等式所获得的界限要好得多。 当 xi=1 时，Van Zuijlen 解决了这个问题。 我们注意到我们的界限在概率论中特别是随机游走理论中有很好的应用。 显示英文摘要

摘要： For a fixed unit vector a=(a_1,a_2,...,a_n) in S^{n-1}, i.e. sum_{i=1}^n a_i^2=1, we consider the 2^n sign vectors epsilon=(epsilon_1,epsilon_2,...,epsilon_n) in {-1,1}^n and the corresponding scalar products a.epsilon=sum_{i=1}^n a_i epsilon_i. Holtzman and Kleitman formulated the following conjecture. It states that among the 2^n sums of the form sum +/- a_i there are not more with |sum_{i=1}^n +/- a_i|>1 than there are with |sum_{i=1}^n +/- a_i| <= 1. The result is of interest in itself, but has also an appealing reformulation in probability theory and in geometry. In this paper we will solve an extension of this problem in the uniform case where all the a's are equal. More precisely, for S_n being a sum of n independent Rademacher random variables, we will give, for several values of xi, precise lower bounds for the probabilities P_n:=P{-xi sqrt{n} <= S_n <= xi sqrt{n}}. There is an obvious relationship with the binomial distribution with parameters n and p=1/2. The obtained lower bounds are sharp and much better than for instance the bound that can be obtained from application of the Chebishev inequality. In case xi=1 Van Zuijlen solved this problem. We remark that our bound will have nice applications in probability theory and especially in random walk theory.