## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 1 (1976), 102-106.

### Nonuniform Central Limit Bounds with Applications to Probabilities of Deviations

#### Abstract

For the distribution of the standardized sum of independent and identically distributed random variables, nonuniform central limit bounds are proved under an appropriate moment condition. From these theorems a condition on the sequence $t_n, n \in \mathbb{N}$, is derived which implies that $1 - F_n(t_n)$ is equivalent to the corresponding deviation of a normally distributed random variable. Furthermore, a necessary and sufficient condition is given for $1 - F_n(t_n) = o(n^{-c/2}t_n^{2 + c})$.

#### Article information

**Source**

Ann. Probab. Volume 4, Number 1 (1976), 102-106.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996186

**Digital Object Identifier**

doi:10.1214/aop/1176996186

**Mathematical Reviews number (MathSciNet)**

MR391226

**Zentralblatt MATH identifier**

0337.60026

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F99: None of the above, but in this section

**Keywords**

Moment conditions approximation central limit theorem deviations

#### Citation

Michel, R. Nonuniform Central Limit Bounds with Applications to Probabilities of Deviations. Ann. Probab. 4 (1976), no. 1, 102--106. doi:10.1214/aop/1176996186. https://projecteuclid.org/euclid.aop/1176996186