Large deviation probability and local density of sets
Abstract
Let $X_1, X_2, \ldots , X_n$ be $n$ independent identically distributed real random variables and $S_n := \displaystyle \sum^n_{i=1} X_i$. We obtain precise asymptotics for $P(S_n \in n A)$ for rather arbitrary Borel sets $A$, in terms of the density of the dominating points in $A$. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics for $P(S_n \in \gamma_n A)$, with $\gamma_n/n \rightarrow \infty$.