小偏差与大偏差之间的一个最优关系（英文）

An Optimal Relation Between Small Deviations and Large Deviations

对于一个取值于Banach空间的高斯随机变量,Gao和Yang最近在[Sci.China Math.,2015,58(5):1091-1100]中推导出小偏差(小球概率)与大偏差(大球概率)之间的一个关系.本文证明了这个关系在一般意义上是最优的.证明的方法是构造一个取值于某个Banach空间的高斯随机变量.本文还考虑了关联于这个关系的几个例子.

For a centered Gaussian random variable in a Banach space, Gao and Yang in [Sci. China Math., 2015, 58（5）： 1091-1100] recently obtained a direct relation between small deviations （small ball probabilities） and large deviations （large ball probabilities）. In this note, we prove that such a relation is optimal in general by carefully selecting a Gaussian random variable in a suitable Banach space. Several examples are considered as well regarding the relation.