Brazilian Journal of Probability and Statistics

Precise asymptotics for products of sums and U-statistics

Zhongquan Tan

Full-text: Open access

Abstract

Let $\{X,X_{i},i\geq 1\}$ be a sequence of independent and identically distributed positive random variables with $E(X)=\mu >0$, $\operatorname{Var}(X)<\infty$. Put $S_{n}=\sum_{i=1}^{n}X_{i}$ and let $g(x)$ be a positive and differentiable function defined on $(0,+\infty)$ satisfying some mild conditions. We prove that, for any $s>1$, \[\lim_{\varepsilon\rightarrow0}\varepsilon^{1/s}\sum_{n=1}^{\infty}g'(n)P\Biggl\{\Biggl|\log\Biggl(\prod_{j=1}^{n}\frac{S_{j}}{j\mu}\Biggr)\Biggr|\geq\varepsilon\sqrt{n}g^{s}(n)\Biggr\}=E|N|^{1/s},\] where $N$ is a standard normal random variable. This result was also extended to product of U-statistics.

Article information

Source
Braz. J. Probab. Stat., Volume 27, Number 1 (2013), 20-30.

Dates
First available in Project Euclid: 16 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1350394627

Digital Object Identifier
doi:10.1214/11-BJPS146

Mathematical Reviews number (MathSciNet)
MR2991776

Zentralblatt MATH identifier
1319.60062

Keywords
Precise asymptotics product of partial sums U-statistics

Citation

Tan, Zhongquan. Precise asymptotics for products of sums and U-statistics. Braz. J. Probab. Stat. 27 (2013), no. 1, 20--30. doi:10.1214/11-BJPS146. https://projecteuclid.org/euclid.bjps/1350394627


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