The Annals of Probability

Self-normalized Cramér-type large deviations for independent random variables

Bing-Yi Jing, Qi-Man Shao, and Qiying Wang

Full-text: Open access

Abstract

Let $X_1, X_2, \ldots $ be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramér-type large deviation result for the standardized partial sums. In this paper, we show that a Cramér-type large deviation theorem holds for self-normalized sums only under a finite $(2+\delta)$th moment, $0< \delta \leq 1$. In particular, we show $P(S_n /V_n \geq x)=\break (1-\Phi(x)) (1+O(1) (1+x)^{2+\delta} /d_{n,\delta}^{2+\delta})$ for $0 \leq x \leq d_{n,\delta}$,\vspace{1pt} where $d_{n,\delta} = (\sum_{i=1}^n EX_i^2)^{1/2}/(\sum_{i=1}^n E|X_i|^{2+\delta})^{1/(2+\delta)}$ and $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.

Article information

Source
Ann. Probab., Volume 31, Number 4 (2003), 2167-2215.

Dates
First available in Project Euclid: 12 November 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1068646382

Digital Object Identifier
doi:10.1214/aop/1068646382

Mathematical Reviews number (MathSciNet)
MR2016616

Zentralblatt MATH identifier
1051.60031

Subjects
Primary: 60F10: Large deviations 60F15: Strong theorems
Secondary: 60G50: Sums of independent random variables; random walks 62F03: Hypothesis testing

Keywords
Large deviation moderate deviation nonuniform Berry--Esseen bound self-normalized sum t-statistic law of the iterated logarithm Studentized bootstrap

Citation

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. doi:10.1214/aop/1068646382. https://projecteuclid.org/euclid.aop/1068646382


Export citation

References

  • Alberink, I. B. (2000). A Berry--Esseen bound for U-statistics in the non-iid case. J. Theorect. Probab. 13 519--533.
  • Bentkus, V. (1994). On the asymptotical behavior of the constant in the Berry--Esseen inequality. J. Theoret. Probab. 7 211--224.
  • Bentkus, V., Bloznelis, M. and Götze, F. (1996). A Berry--Esseen bound for Student's statistic in the non-i.i.d. case. J. Theoret. Probab. 9 765--796.
  • Bentkus, V. and Götze, F. (1996). The Berry--Esseen bound for Student's statistic. Ann. Prob. 24 491--503.
  • Bentkus, V., Götze, F. and van Zwet, W. R. (1997). An Edgeworth expansion for symmetric statistics. Ann. Statist. 25 851--896.
  • Chistyakov, G. P. and Götze, F. (1999). Moderate deviations for self-normalized sums. Preprint.
  • Csörgő, S., Haeusler, E. and Mason, D. M. (1988). The asymptotic distribution of trimmed sums. Ann. Probab. 16 672--699.
  • Csörgő, S., Haeusler, E. and Mason, D. M. (1991). The quantile-transform approach to the asymptotic distribution of modulus trimmed sums. In Sums, Trimmed Sums and Extremes (M. G. Hahn, D. M. Mason and D. C. Weiner, eds.) 337--354. Birkhäuser, Boston.
  • Csörgő, M. and Horváth, L. (1988). Asymptotic representations of self-normalized sums. Probab. Math. Statist. 9 15--24.
  • Daniels, H. E. and Young, G. A. (1991). Saddlepoint approximation for the Studentized mean, with an application to the bootstrap. Biometrika 78 169--179.
  • Darling, D. A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95--107.
  • Davison, A. C. and Hinkley, D. V. (1997). Bootstrap Methods and Their Application. Cambridge Univ. Press.
  • Dharmadhikari, S. W., Fabian, V. and Jogdeo, K. (1968). Bounds on the moments of martingales. Ann. Math. Statist. 39 1719--1723.
  • Efron, B. (1979). Bootstrap methods: Another look at the jackknife. Ann. Statist. 7 1--26.
  • Efron, B. and Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Chapman and Hall, New York.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Wiley, New York.
  • Giné, E., Götze, F. and Mason, D. M. (1997). When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 1514--1531.
  • Griffin, P. S. and Kuelbs, J. D. (1989). Self-normalized laws of the iterated logarithm. Ann. Probab. 17 1571--1601.
  • Griffin, P. S. and Mason, D. M. (1991). On the asymptotic normality of self-normalized sums. Math. Proc. Camb. Phil. Soc. 109 597--610.
  • Griffin, P. S. and Pruitt, W. E. (1989). Asymptotic normality and subsequential limits of trimmed sums. Ann. Probab. 17 1186--1219.
  • Hahn, M. G., Kuelbs, J. and Weiner, D. C. (1990). The asymptotic joint distribution of self-normalized censored sums and sums of squares. Ann. Probab. 18 1284--1341.
  • Hahn, M. G., Kuelbs, J. and Weiner, D. C. (1991). Asymptotic behavior of partial sums: A more robust approach via trimming and self-normalization. In Sums, Trimmed Sums and Extremes (M. G. Hahn, D. M. Mason and D. C. Weiner, eds.) 1--54. Birkhäuser, Boston.
  • Hall, P. (1988). Theorectical comparison of bootstrap confidence intervals (with discussion). Ann. Statist. 35 108--129.
  • Hall, P. (1990). On the relative performance of bootstrap and Edgeworth approximations of a distribution function. J. Multivariate Anal. 35 108--129.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • He, X. and Shao, Q. M. (2000). On parameters of increasing dimensions. J. Multivariate Anal. 73 120--135.
  • Jing, B.-Y., Feuerverger, A. and Robinson, J. (1994). On the bootstrap saddlepoint approximations. Biometrika 81 211--215.
  • Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, New York.
  • LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624--632.
  • Logan, B. F., Mallows, C. L., Rice, S. O. and Shepp, L. A. (1973). Limit distributions of self-normalized sums. Ann. Probab. 1 788--809.
  • Maller, R. (1988). Asymptotic normality of trimmed means in higher dimensions. Ann. Probab. 16 1608--1622.
  • Maller, R. (1991). A review of some asymptotic properties of trimmed sums of multivariate data. In Sums, Trimmed Sums and Extremes (M. G. Hahn, D. M. Mason and D. C. Weiner, eds.) 179--214. Birkhäuser, Boston.
  • Petrov, V. V. (1965). On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10 287--298.
  • Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • Prawitz, H. (1972). Limits for a distribution, if the characteristic function is given in a finite domain. Skand. Aktuar. Tidskr. 138--154.
  • Shao, Q.-M. (1995). Strong approximation theorems for independent random variables and their applications. J. Multivariate Anal. 52 107--130.
  • Shao, Q.-M. (1997). Self-normalized large deviations. Ann. Probab. 25 285--328.
  • Shao, Q.-M. (1998). Recent developments in self-normalized limit theorems. In Asymptotic Methods in Probability and Statistics (B. Szyszkowicz, ed.) 467--480. Birkhäuser, Boston.
  • Shao, Q.-M. (1999). Cramér-type large deviation for Student's $t$ statistic. J. Theoret. Probab. 12 387--398.
  • Singh, K. (1981). On the asymptotic accuracy of Efron's bootstrap. Ann. Statist. 9 1187--1195.
  • Strassen, V. (1966). A converse to the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 4 265--268.
  • Wang, Q. and Jing, B.-Y. (1999). An exponential non-uniform Berry--Esseen bound for self-normalized sums. Ann. Probab. 27 2068--2088.
  • Wittmann, R. (1987). Sufficient moment and truncated moment conditions for the law of the iterated logarithm. Probab. Theory Related Fields 75 509--530.
  • Wood, A. T. A. (2000). Bootstrap relative errors and sub-exponential distributions. Bernoulli 6 809--834.