论文标题
Cramer定理的亚指数版本
A subexponential version of Cramer's theorem
论文作者
论文摘要
我们考虑与次指数矩条件下自变和相同分布的随机变量相关的大偏差。我们表明，在变量数量中可以在次指数尺度上观察到非平凡的偏差，并且我们提供了相关的速率函数，该函数是非凸的，并且不是源自Legendre-Fenchel变换。证明将Cramer定理之一调整为单个变量产生波动的情况。特别是，我们为下边界制定了一种新的倾斜策略，这使我们引入了对当时生成函数的第二个导数的条件。我们的结果通过几个简单的示例来说明。
We consider the large deviations associated with the empirical mean of independent and identically distributed random variables under a subexponential moment condition. We show that non-trivial deviations are observable at a subexponential scale in the number of variables, and we provide the associated rate function, which is non-convex and is not derived from a Legendre-Fenchel transform. The proof adapts the one of Cramer's theorem to the case where the fluctuation is generated by a single variable. In particular, we develop a new tilting strategy for the lower bound, which leads us to introduce a condition on the second derivative of the moment generating function. Our results are illustrated by a couple of simple examples.
