The Annals of Probability

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

Philip S. Griffin

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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


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