# 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 $\gamma_n/n \rightarrow \infty$.
keyword :
Document type :
Reports
Domain :

https://hal.inria.fr/inria-00074057
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 2:24:13 PM
Last modification on : Friday, May 25, 2018 - 12:02:05 PM
Long-term archiving on: : Sunday, April 4, 2010 - 10:11:08 PM

### Identifiers

• HAL Id : inria-00074057, version 1

### Citation

Philippe Barbe, Michel Broniatowski. Large deviation probability and local density of sets. [Research Report] RR-2630, INRIA. 1995. ⟨inria-00074057⟩

Record views