与极值组合学问题有关的随机游动的最优概率不等式
Optimal Probability Inequalities for Random Walks related to Problems in Extremal Combinatorics
摘要：
设S_n=X_1+…+ X_n是独立对称随机变量之和，使得|X_{i}| \leq 1.记为W_n=ε_{1}+.+ε_{n}是满足\prob{\eps_i = \pm 1} = 1/2的独立随机变量之和。我们证明了\mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}，其中A是一个区间的形式[x，\infty）或只是一个单一的点。该不等式是精确的，并明确给出了c和k的最优值。在Rademacher级数的情况下，它改进了Kwapiestan不等式。我们还提供了一个新的和非常短的证明Littlewood-Offord问题，而不使用Sperner定理。最后，给出了奇Lipschitz函数的一个推广.
英文摘要：
Let S_n=X_1+...+X_n be a sum of independent symmetric random variables such that |X_{i}|\leq 1. Denote by W_n=ε_{1}+...+ε_{n} a sum of independent random variables such that \prob{\eps_i = \pm 1} = 1/2. We prove that \mathbb{P}{S_{n} \in A} \leq \mathbb{P}{cW_k \in A}, where A is either an interval of the form [x, \infty) or just a single point. The inequality is exact and the optimal values of c and k are given explicitly. It improves Kwapień's inequality in the case of the Rademacher series. We also provide a new and very short proof of the Littlewood-Offord problem without using Sperner's Theorem. Finally, an extension to odd Lipschitz functions is given.
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