## Electronic Journal of Probability

### Asymptotic Distributions and Berry-Esseen Bounds for Sums of Record Values

#### Abstract

Let $\{U_n, n \geq 1\}$ be independent uniformly distributed random variables, and $\{Y_n, n \geq 1\}$ be independent and identically distributed non-negative random variables with finite third moments. Denote $S_n = \sum_{i=1}^n Y_i$ and assume that $(U_1, \cdots, U_n)$ and $S_{n+1}$ are independent for every fixed $n$. In this paper we obtain Berry-Esseen bounds for $\sum_{i=1}^n \psi(U_i S_{n+1})$, where $\psi$ is a non-negative function. As an application, we give Berry-Esseen bounds and asymptotic distributions for sums of record values.

#### Article information

Source
Electron. J. Probab., Volume 9 (2004), paper no. 17, 544-559.

Dates
Accepted: 9 June 2004
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ejp/1465229703

Digital Object Identifier
doi:10.1214/EJP.v9-210

Mathematical Reviews number (MathSciNet)
MR2080608

Zentralblatt MATH identifier
1064.60047

Subjects
Primary: 60F10: Large deviations
Secondary: 60F15: Strong theorems

Rights

#### Citation

Shao, Qi-Man; Su, Chun; Wei, Gang. Asymptotic Distributions and Berry-Esseen Bounds for Sums of Record Values. Electron. J. Probab. 9 (2004), paper no. 17, 544--559. doi:10.1214/EJP.v9-210. https://projecteuclid.org/euclid.ejp/1465229703

#### References

• Arnold, B.C. and Villasenor, J.A. (1998). The asymptotic distributions of sums of records, Extremes 1, 351-363.
• Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Cambridge University Press, Cambridge.
• Chen, L.H.Y. and Shao, Q.M. (2003). Uniform and non-uniform bounds in normal approximation for nonlinear Statistics. Preprint.
• de Haan, L and Resnick, S.I. (1973). Almost sure limit points of record values. J. Appl. Probab. 10, 528–542.
• Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
• Hu, Z.S., Su, C. and Wang, D.C. (2002). The asymptotic distributions of sums of record values for distributions with lognormal-type tails. Sci. China Ser. A 45, 1557–1566.
• Mikosch, T. and Nagaev, A.V. (1998). Large Deviations of heavy-tailed sums with applications in insurance. Extremes 1, 81-110.
• Petrov, V.V. (1995). Limit Theorems of Probability Theory, Sequences of Independent Random Variables. Clarendon Press, Oxford.
• Resnick, S.I. (1973). Limit laws for record values. Stoch. Process. Appl. 1, 67-82.
• Su, C. and Hu, Z.S. (2002). The asymptotic distributions of sums of record values for distributions with regularly varying tails. J. Math. Sci. (New York) 111, 3888–3894.
• Tata, M.N. (1969). On outstanding values in a sequence of random variables. Z. Wahrsch. Verw. Gebiete 12, 1969 9–20.