Electronic Journal of Probability

Limit Theorems for Self-Normalized Large Deviation

Qiying Wang

Full-text: Open access

Abstract

Let $X, X_1, X_2, \cdots $ be i.i.d. random variables with zero mean and finite variance $\sigma^2$. It is well known that a finite exponential moment assumption is necessary to study limit theorems for large deviation for the standardized partial sums. In this paper, limit theorems for large deviation for self-normalized sums are derived only under finite moment conditions. In particular, we show that, if $EX^4 < \infty$, then $$ \frac{P(S_n /V_n \geq x)}{1-\Phi(x)} \exp\left\{ -\frac{x^3 EX^3}{3\sqrt{ n}\sigma^3} \right\} \left[ 1 + O\left(\frac{1+x} {\sqrt {n}}\right) \right], $$ for $x \ge 0$and $x = O(n^{1/6})$, where $S_n\sum_{i=1}^nX_i$ and $V_n (\sum_{i=1}^n X_i^2)^{1/2}$.

Article information

Source
Electron. J. Probab., Volume 10 (2005), paper no. 38, 1260-1285.

Dates
Accepted: 14 November 2005
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464816839

Digital Object Identifier
doi:10.1214/EJP.v10-289

Mathematical Reviews number (MathSciNet)
MR2176384

Zentralblatt MATH identifier
1112.60020

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 62E20: Asymptotic distribution theory

Keywords
Cram'er large deviation limit theorem

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Wang, Qiying. Limit Theorems for Self-Normalized Large Deviation. Electron. J. Probab. 10 (2005), paper no. 38, 1260--1285. doi:10.1214/EJP.v10-289. https://projecteuclid.org/euclid.ejp/1464816839


Export citation

References

  • Bentkus, V.; Bloznelis, M.; Götze, F. A Berry-Esséen bound for Student's statistic in the non-i.i.d. case. J. Theoret. Probab. 9 (1996), no. 3, 765–796.
  • Bentkus, V.; Götze, F.. The Berry-Esseen bound for Student's statistic. Ann. Probab. 24 (1996), 491–503.
  • Bentkus, V.; Götze, F.; van Zwet, W. R. An Edgeworth expansion for symmetric statistics. Ann. Statist. 25 (1997), no. 2, 851–896.
  • Chistyakov, G. P.; Götze, F. Moderate deviations for Student's statistic. Teor. Veroyatnost. i Primenen. 47 (2003), no. 3, 415–428.
  • Friedrich, Karl O. A Berry-Esseen bound for functions of independent random variables. Ann. Statist. 17 (1989), no. 1, 170–183.
  • Giné, Evarist; Götze, Friedrich; Mason,David M. When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 (1997), no. 3, 1514–1531.
  • Hall, Peter. Edgeworth expansion for Student's $t$ statistic under minimal moment conditions. Ann. Probab. 15 (1987), no. 3, 920–931.
  • Hall, Peter. On the effect of random norming on the rate of convergence in the central limit theorem. Ann. Probab. 16 (1988), no. 3, 1265–1280.
  • Hall, Peter; Jing, Bing-Yi. Uniform coverage bounds for confidence intervals and Berry-Esseen theorems for Edgeworth expansion. Ann. Statist. 23 (1995), no. 2, 363–375.
  • He, Xuming; Shao, Qi-Man. On parameters of increasing dimensions. J. Multivariate Anal. 73 (2000), no. 1, 120–135.
  • Jing, Bing-Yi; Shao, Qi-Man; Wang, Qiying. Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 (2003), no. 4, 2167–2215.
  • Logan, B. F.; Mallows, C. L.; Rice, S. O.; Shepp, L. A. Limit distributions of self-normalized sums. Ann. Probability 1 (1973), 788–809. (50 #14890)
  • Petrov, Valentin V. Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability, 4. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.
  • Prawitz, H.akan. Limits for a distribution, if the characteristic function is given in a finite domain. Skand. Aktuarietidskr. 1972, 138–154 (1973) (51 #11622)
  • Putter, Hein; van Zwet, Willem R. Empirical Edgeworth expansions for symmetric statistics. Ann. Statist. 26 (1998), no. 4, 1540–1569.
  • Slavova, V. V. On the Berry-Esseen bound for Student's statistic. Stability problems for stochastic models (Uzhgorod, 1984), 355–390, Lecture Notes in Math., 1155, Springer, Berlin, 1985.
  • Shao, Qi-Man. A Cramér type large deviation result for Student's $t$-statistic. J. Theoret. Probab. 12 (1999), no. 2, 385–398.
  • van Zwet, W. R. A Berry-Esseen bound for symmetric statistics. Z. Wahrsch. Verw. Gebiete 66 (1984), no. 3, 425–440.
  • Wang, Qiying; Jing, Bing-Yi. An exponential nonuniform Berry-Esseen bound for self-normalized sums. Ann. Probab. 27 (1999), no. 4, 2068–2088.
  • Wang, Qiying; Jing, Bing-Yi; Zhao, Lincheng. The Berry-Esseen bound for Studentized statistics. Ann. Probab. 28 (2000), no. 1, 511–535.