## Brazilian Journal of Probability and Statistics

### Precise asymptotics for products of sums and U-statistics

Zhongquan Tan

#### 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

https://projecteuclid.org/euclid.bjps/1350394627

Digital Object Identifier
doi:10.1214/11-BJPS146

Mathematical Reviews number (MathSciNet)
MR2991776

Zentralblatt MATH identifier
1319.60062

#### 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

#### References

• Arnold, B. C. and Villasenor, J. A. (1998). The asymptotic distribution of sums of records. Extremes 1, 351–363.
• Baum, L. E. and Katz, M. (1965). Convergence rates in the law of large numbers. Transactions of the American Mathematical Society 120, 108–123.
• Cheng, F. Y. and Wang, Y. B. (2004). Precise asymptotics of partial sums for IID and NA sequences (in Chinese). Acta Mathematica Sinica, Ser. A 45, 965–972.
• Davis, J. A. (1968). Convergence rates for the law of the iterated logarithm. Annals of Mathematical Statistics 39, 1479–1485.
• Gonchigdanzan, K. and Rempala, G. (2006). A note on the almost sure limit theorem for the product of partial sums. Applied Mathematics Letters 19, 191–196.
• Gut, A. (2002). Precise asymptotics for record times and the associated counting process. Stochastic Processes and Their Applications 101, 233–239.
• Gut, A. and Spătaru, A. (2000a). Precise asymptotics in the Baum–Katz and Davis law of large numbers. Journal of Mathematical Analysis and Applications 248, 233–246.
• Gut, A. and Spătaru, A. (2000b). Precise asymptotics in the law of the iterated logarithm. The Annals of Probability 28, 1870–1883.
• Heyde, C. C. (1975). A supplement to the strong law of large numbers. Journal of Applied Probability 12, 173–175.
• Lu, X. and Qi, Y. (2004). A note on asymptotic distribution of products of sums. Statistics and Probability Letters 68, 407–413.
• Matula, P. and Stẹpien, I. (2008). On the application of strong approximation to weak convergence of products of sums for dependent random variables. Condensed Matter Physics 11, 749–754.
• Matula, P. and Stẹpien, I. (2009). Weak convergence of products of sums of independent and non-identically distributed random variables. Journal of Mathematical Analysis and Applications 335, 45–54.
• Qi, Y. (2003). Limit distributions for products of sums. Statistics and Probability Letters 62, 93–100.
• Rempala, G. and Wesolowski, J. (2002). Asymptotics for products of sums and U-statistics. Electronic Communications in Probability 7, 47–54.
• Rempala, G. and Wesolowski, J. (2005). Asymptotics for products of independent sums with an application to Wishart determinants. Statistics and Probability Letters 74, 129–138.
• Serfling, W. (1980). Approximation Theorems of Mathematical Statistics. New York: Wiley.
• Wang, Y. B. and Yang, Y. (2003). A general law of precise asymptotics for the counting process of record times. Journal of Mathematical Analysis and Applications 286, 753–764.
• Zhang, L. and Huang, W. (2007). A note on the invariance principle of the product of sums of random variables. Electronic Communications in Probability 12, 51–56.