## Electronic Journal of Probability

### Cramér Type Moderate deviations for the Maximum of Self-normalized Sums

#### Abstract

Let $\{ X, X_i , i \geq 1\}$ be i.i.d. random variables, $S_k$ be the partial sum and $V_n^2 = \sum_{1\leq i\leq n} X_i^2$. Assume that $E(X)=0$ and $E(X^4) < \infty$. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that $P(\max_{1 \leq k \leq n} S_k \geq x V_n) / (1- \Phi(x)) \to 2$ uniformly in $x \in [0, o(n^{1/6}))$.

#### Article information

Source
Electron. J. Probab., Volume 14 (2009), paper no. 41, 1181-1197.

Dates
Accepted: 31 May 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819502

Digital Object Identifier
doi:10.1214/EJP.v14-663

Mathematical Reviews number (MathSciNet)
MR2511281

Zentralblatt MATH identifier
1272.68116

Subjects
Primary: 60F10: Large deviations
Secondary: 62E20: Asymptotic distribution theory

Rights

#### Citation

Hu, Zhishui; Shao, Qi-Man; Wang, Qiying. Cramér Type Moderate deviations for the Maximum of Self-normalized Sums. Electron. J. Probab. 14 (2009), paper no. 41, 1181--1197. doi:10.1214/EJP.v14-663. https://projecteuclid.org/euclid.ejp/1464819502

#### References

• Aleshkyavichene, A. K. (1979). Probabilities of large deviations for the maximum of sums of independent random variables. Theory Prob. Applications 24 16–33. (80j:60038a)
• Aleshkyavichene, A. K. (1979). Probabilities of large deviations for the maximum of sums of independent random variables. II. Theory Prob. Applications 24, 322-337. (80j:60038b)
• Arak, T.V. (1974). On the distribution of the maximum of successive partial sums of independent random variables. Theory Probab. Appl. 19 245-266.
• Arak, T.V. and Nevzorov, V.B. (1973). Some estimates for the maximum cumulative sum of independent random variables. Theory Probab. Appl. 18 384-387.
• Chistyakov, G. P. and Götze, F. (2004). Limit distributions of Studentized means. Ann. Probab. 32 no. 1A, 28-77.
• Csörgó, M., Szyszkowicz, B. and Wang, Q. (2003a). Darling-Erdös theorem for self-normalized sums. Ann. Probab. 31 676-692.
• Csörgó, M., Szyszkowicz, B. and Wang, Q. (2003b). Donsker's theorem for self-normalized partial sums processes. Ann. Probab. 31 1228–1240.
• Giné, E., Götze, F. and Mason, D. (1997). When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 1514–1531.
• Griffin, P. and Kuelbs, J. (1989) Self-normalized laws of the iterated logarithm. Ann. Probab. 17 1571–1601.
• Hall, P. and Wang, Q. (2004). Exact convergence rate and leading term in central limit theorem for student's $t$ statistic. Ann. Probab. 32 1419-1437.
• Jing, B.-Y. Shao, Q.-M. and Wang, Q. (2003). Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 2167-2215.
• Jing, B.-Y., Shao, Q.-M. and Zhou, W. (2004). Saddlepoint approximation for Student's $t$-statistic with no moment conditions. Ann. Statist. 32 2679-2711.
• Nagaev, S. V. (1969). An estimate of the rate of convergence of the distribution of the maximum of the sums of independent random variables. Siberian Math. J. 10 614–633 (in Russian).
• Nevzorov, V. B. (1973). On the distribution of the maximum sum of independent terms. Soviet Math. Dolk. 14 40–42.
• Robinson, J. and Wang, Q. (2005). On the self-normalized Cramér-type large deviation. Journal of Theoret. Probab. 18 891-909.
• Sakhanenko, A. I.(1992). Berry-Esseen type estimates for large deviation probabilities. Siberian Math. J. 32 647–656.
• Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285–328..
• Shao, Q.-M. (1999). A Cramer type large deviation result for Student's $t$-statistic. J. Theoret. Probab. 12 385–398.
• Wang, Q. (2005). Limit Theorems for self-normalized large deviation. Electronic Journal of Probab. 10 1260-1285.
• Wang, Q. and Jing, B.-Y. (1999). An exponential nonuniform Berry-Esseen bound for self-normalized sums. Ann. Probab. 27 2068–2088.
• Zhou, W. and Jing, B.-Y. (2006). Tail probability approximations for Student's $t$ statistics. Probab. Theory Relat. Fields 136 541-559.