The Annals of Probability

On the Upper Bound for Large Deviations of Sums of I.I.D. Random Vectors

M. Slaby

Full-text: Open access

Abstract

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random vectors with values in $\mathbb{R}^d, \mu = \mathscr{L}(X_1)$ and let $\lambda$ be the convex conjugate of $\log \hat{\mu}$, where $\hat{\mu}$ is the Laplace transform of $\mu$. For every $d \geq 2$, a probability measure $\mu$ and an open set $A$ in $\mathbb{R}^d$ are constructed so that $\lim \inf_{n\rightarrow\infty} \frac{1}{n} \log P\big(\frac{S_n}{n} \in A\big) > - \Lambda (A),$ where $S_n = X_1 + \cdots + X_n$ and $\Lambda (A) = \inf_{x \in A} \Lambda (x)$. It is also shown that if $\mu$ satisfies certain regularity conditions, then $\lim \sup_{n\rightarrow \infty} \frac{1}{n} \log P\big(\frac{S_n}{n} \in A\big) \leq - \Lambda (A),$ holds for all Borel sets in $\mathbb{R}^d$.

Article information

Source
Ann. Probab., Volume 16, Number 3 (1988), 978-990.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991672

Digital Object Identifier
doi:10.1214/aop/1176991672

Mathematical Reviews number (MathSciNet)
MR942750

Zentralblatt MATH identifier
0649.60034

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations

Keywords
Sums of i.i.d. random vectors large deviations upper bound for open sets

Citation

Slaby, M. On the Upper Bound for Large Deviations of Sums of I.I.D. Random Vectors. Ann. Probab. 16 (1988), no. 3, 978--990. doi:10.1214/aop/1176991672. https://projecteuclid.org/euclid.aop/1176991672


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