## The Annals of Probability

- Ann. Probab.
- Volume 11, Number 4 (1983), 939-952.

### Probability Estimates for the Small Deviations of $d$-Dimensional Random Walk

#### Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent, identically distributed random variables taking values in $\mathbb{R}^d$ and $S_n = X_1 + \cdots + X_n$. For a large class of distributions we obtain estimates for the probability that $S_n$ is in a ball centered at the origin. Such an estimate would follow from a local limit theorem if $X_1$ were in the domain of attraction of a stable law.

#### Article information

**Source**

Ann. Probab., Volume 11, Number 4 (1983), 939-952.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993443

**Digital Object Identifier**

doi:10.1214/aop/1176993443

**Mathematical Reviews number (MathSciNet)**

MR714957

**Zentralblatt MATH identifier**

0519.60044

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60E15: Inequalities; stochastic orderings

**Keywords**

Probability estimate local limit theorem

#### Citation

Griffin, Philip S. Probability Estimates for the Small Deviations of $d$-Dimensional Random Walk. Ann. Probab. 11 (1983), no. 4, 939--952. doi:10.1214/aop/1176993443. https://projecteuclid.org/euclid.aop/1176993443