 |
|
|
| | | | | |
|
|
|
|
|
Cram'er Type Moderate deviations for the Maximum of Self-normalized Sums
|
Zhishui Hu, USTC
Qi-Man Shao, HKUST
Qiying Wang, University of Sydney
|
Abstract
Let { X, Xi , i ≥ 1} be i.i.d. random variables, Sk be the partial sum and Vn2
= ∑1&le i&le n Xi2. Assume that E(X)=0 and E(X4) < ∞. In this paper we discuss the moderate deviations of the maximum of the self-normalized sums. In particular, we prove that P(max1 ≤ k ≤ n Sk &ge x Vn) / (1- Φ(x)) → 2 uniformly in x ∈ [0, o(n1/6)).
|
Full text: PDF
Pages: 1181-1197
Published on: May 31, 2009
|
Bibliography
-
Aleshkyavichene, A. K. (1979).
Probabilities of large deviations for the maximum of sums of independent random variables.
Theory Prob. Applications 24 16--33.
MR0522234 (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.
MR0532445 (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.
MR0343344 (49 #8086)
-
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.
MR0341563 (49 #6311)
-
Chistyakov, G. P. and G"otze, F. (2004).
Limit distributions of Studentized means. Ann. Probab. 32 no. 1A, 28-77.
MR2040775 (2005f:60055)
-
Cs"orgH{o}, M., Szyszkowicz, B. and Wang, Q. (2003a).
Darling-Erd"os theorem for self-normalized sums.
Ann. Probab. 31 676-692.
MR1964945 (2004a:60051)
-
Cs"orgH{o}, M., Szyszkowicz, B. and Wang, Q. (2003b).
Donsker's theorem for self-normalized partial sums processes.
Ann. Probab. 31 1228--1240.
MR1988470 (2004h:60031)
-
Gin'e, E., G"otze, F. and Mason, D. (1997).
When is the Student $t$-statistic asymptotically standard normal? Ann. Probab. 25 1514--1531.
MR1457629 (98j:60033)
-
Griffin, P. and Kuelbs, J. (1989) Self-normalized laws of the
iterated logarithm. Ann. Probab. 17 1571--1601.
MR1048947 (91k:60036)
-
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.
MR2060303 (2005e:62025)
-
Jing, B.-Y. Shao, Q.-M. and Wang, Q. (2003).
Self-normalized Cram'er-type large deviations for independent random variables.
Ann. Probab. 31 2167-2215.
MR2016616 (2004k:60069)
-
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.
MR2153999 (2006a:62030)
-
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).
MR0245075(39 #6387)
-
Nevzorov, V. B. (1973). On the distribution of the maximum sum of independent terms.
Soviet Math. Dolk. 14 40--42.
MR0315767(47 #4316)
-
Robinson, J. and Wang, Q. (2005).
On the self-normalized Cram'er-type large deviation.
Journal
of Theoret. Probab. 18 891-909.
MR2300002 (2008h:60090)
-
Sakhanenko, A. I.(1992). Berry-Esseen type estimates
for large deviation probabilities.
Siberian Math. J. 32 647--656.
MR1142075 (93e:60054)
-
Shao, Q.-M. (1997). Self-normalized large deviations.
Ann. Probab. 25 285--328.
MR1428510 (98b:60056)
.
-
Shao, Q.-M. (1999). A Cramer type large deviation result
for Student's $t$-statistic.
J. Theoret. Probab. 12 385--398.
MR1684750 (2000d:60046)
-
Wang, Q. (2005). Limit Theorems for self-normalized large deviation.
Electronic Journal of Probab. 10 1260-1285.
MR2176384 (2006j:60029)
-
Wang, Q. and Jing, B.-Y. (1999). An exponential nonuniform
Berry-Esseen bound for self-normalized sums. Ann. Probab. 27 2068--2088.
MR1742902 (2001c:60045)
-
Zhou, W. and Jing, B.-Y. (2006). Tail probability approximations for
Student's $t$ statistics. Probab. Theory Relat. Fields 136 541-559.
MR2257135 (2007k:62034)
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context