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February 2013 Precise asymptotics for products of sums and U-statistics
Zhongquan Tan
Braz. J. Probab. Stat. 27(1): 20-30 (February 2013). DOI: 10.1214/11-BJPS146

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.

Citation

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Zhongquan Tan. "Precise asymptotics for products of sums and U-statistics." Braz. J. Probab. Stat. 27 (1) 20 - 30, February 2013. https://doi.org/10.1214/11-BJPS146

Information

Published: February 2013
First available in Project Euclid: 16 October 2012

zbMATH: 1319.60062
MathSciNet: MR2991776
Digital Object Identifier: 10.1214/11-BJPS146

Rights: Copyright © 2013 Brazilian Statistical Association

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Vol.27 • No. 1 • February 2013
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