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Towards a universal self-normalized moderate deviation

Author(s): Bing-Yi Jing; Qi-Man Shao; Wang Zhou
Journal: Trans. Amer. Math. Soc. 360 (2008), 4263-4285.
MSC (2000): Primary 60F10, 60F15; Secondary 60G50
Posted: March 20, 2008
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Abstract: This paper is an attempt to establish a universal moderate deviation for self-normalized sums of independent and identically distributed random variables without any moment condition. The exponent term in the moderate deviation is specified when the distribution is in the centered Feller class. An application to the law of the iterated logarithm is given.

References:

1.
V. Bentkus and F. Götze. The Berry-Esseen bound for Student's statistic. Ann. Probab. 24 (1996), 491-503. MR 1387647 (97f:62021)

2.
M. Csörgő, B. Szyszkowicz and Q. Wang. Darling-Erdős theorem for self-normalized sums. Ann. Probab. 31 (2003), 676-692. MR 1964945 (2004a:60051)

3.
M. Csörgő, B. Szyszkowicz and Q. Wang. Donsker's theorem for self-normalized partial sums processes. Ann. Probab. 31 (2003), 1228-1240. MR 1988470 (2004h:60031)

4.
R.M. Dudley. Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, Vol. 63. Cambridge University Press, Cambridge, 1999. MR 1720712 (2000k:60050)

5.
U. Einmahl and D.M. Mason. Some universal results on the behavior of increments of partial sums. Ann. Probab. 24 (1996), 1388-1407. MR 1411499 (97m:60034)

6.
E. Giné, F. Götze and D. M. Mason. When is the Student $ t$-statistic asymptotically standard normal? Ann. Probab. 25 (1997), 1514-1531. MR 1457629 (98j:60033)

7.
E. Giné and D. M. Mason. On the LIL for self-normalized sums of IID random variables. J. Theoret. Probab. 11 (1998), 351-370. MR 1622575 (99e:60082)

8.
F. Götze and G. P. Chistyakov. Limit distributions of studentized means. Ann. Probab. 32 (2004), 28-77. MR 2040775 (2005f:60055)

9.
P. S. Griffin. Tightness of the Student $ t$-statistic. Electron. Comm. Prob. 7 (2002), 171-180. MR 1937903 (2003i:60037)

10.
P. S. Griffin and J. Kuelbs. Self-normalized laws of the iterated logarithm. Ann. Probab. 17 (1989), 1571-1601. MR 1048947 (91k:60036)

11.
B.-Y. Jing, Q. M. Shao and Q. Wang. Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab. 31 (2003), 2167-2215. MR 2016616 (2004k:60069)

12.
B.-Y. Jing, Q. M. Shao and W. Zhou. Saddlepoint approximation for Student's $ t$-statistic with no moment conditions. Ann. Statist. 32 (2004), 2679-2711. MR 2153999 (2006a:62030)

13.
B. F. Logan, C. L. Mallows, S. O. Rice and L. A. Shepp. Limit distributions of self-normalized sums. Ann. Probab. 1 (1973), 788-809. MR 0362449 (50:14890)

14.
Q. M. Shao. Self-normalized large deviations. Ann. Probab. 25 (1997), 285-328. MR 1428510 (98b:60056)

15.
Q. M. Shao. Recent developments in self-normalized limit theorems. In Asymptotic Methods in Probability and Statistics (editor B. Szyszkowicz), pp. 467 - 480. Elsevier Science, 1998. MR 1661499 (99j:60063)

16.
Q. M. Shao. Recent progress on self-normalized limit theorems. Probability, Finance and Insurance. Edited by T.L. Lai, H. Yang, and S.P. Yung, World Scientific, 2004. MR 2189198 (2006i:60056)

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Additional Information:

Bing-Yi Jing
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong
Email: majing@ust.hk

Qi-Man Shao
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong -- and -- Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Email: maqmshao@ust.hk

Wang Zhou
Affiliation: Department of Statistics and Applied Probability, National University of Singapore, Singapore 117546
Email: stazw@nus.edu.sg

DOI: 10.1090/S0002-9947-08-04402-4
PII: S 0002-9947(08)04402-4
Keywords: Moderate deviation, large deviation, self-normalized sums, the law of the iterated logarithm
Received by editor(s): February 7, 2006
Received by editor(s) in revised form: August 14, 2006
Posted: March 20, 2008
Additional Notes: The first author was supported in part by Hong Kong RGC CERG No. HKUST6117/02P and DAG05/06.SC01
The second author was partially supported by the National Science Foundation under Grant No. DMS-0103487 and HKUST DAG 05/06 Sc27 and RGC CERG No. 602206
The third author was partially supported by the grants R-155-000-035-112 and R-155-050-055-133/101 at the National University of Singapore
Copyright of article: Copyright 2008, American Mathematical Society

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