# A SELF-NORMALIZED LIL FOR CONDITIONALLY TRIMMED SUMS AND CONDITIONALLY CENSORED SUMS

• Pang Tian Xiao ;
• Lin Zheng Yan
• Published : 2006.07.01
• 65 3

#### Abstract

Let {$X,\;X_n;n\;{\geq}\;1$} be a sequence of ${\imath}.{\imath}.d.$ random variables which belong to the attraction of the normal law, and $X^{(1)}_n,...,X^{(n)}_n$ be an arrangement of $X_1,...,X_n$ in decreasing order of magnitude, i.e., $\|X^{(1)}_n\|{\geq}{\cdots}{\geq}\|X^{(n)}_n\|$. Suppose that {${\gamma}_n$} is a sequence of constants satisfying some mild conditions and d'($t_{nk}$) is an appropriate truncation level, where $n_k=[{\beta}^k]\;and\;{\beta}$ is any constant larger than one. Then we show that the conditionally trimmed sums obeys the self-normalized law of the iterated logarithm (LIL). Moreover, the self-normalized LIL for conditionally censored sums is also discussed.

#### Keywords

self-normalized;law of the iterated logarithm;trimmed sums;censored sums;${\imath}.{\imath}.d.$ random variables

#### References

1. V. Bentkus and F. Gotze, The Berry-Esseen bound for Student's statistic, Ann. Probab. 24 (1996), no. 1, 491-503 https://doi.org/10.1214/aop/1042644728
2. M. Csorgo, B. Szyszkowicz, and Q. Y. Wang, Donsker's theorem for self-nor- malized partial sums processes, Ann. Probab. 31 (2003), no. 3, 1228-1240 https://doi.org/10.1214/aop/1055425777
3. P. S. Griffin, Non-classical law of the iterated logarithm behaviour for trimmed sums, Probab. Theory Related Fields 78 (1988), no. 2, 293-319 https://doi.org/10.1007/BF00322025
4. M. Hahn and J. Kuelbs, Universal asymptotic normality for conditionally trimmed sums, Stat. Prob. Lett. 7 (1988), no. 1, 9-15 https://doi.org/10.1016/0167-7152(88)90079-X
5. M. Hahn, J. Kuelbs, and D. C. Weiner, A universal law of the iterated logarithm for trimmed and censored sums, Lecture Notes in Math. 1391, Springer, Berlin, 1989
6. Z. Y. Lin, The law of the iterated logarithm for the rescaled R=S statistics without the second moment, J. Compt. Math. Appl. 47 (2004), no. 8-9, 1389-1396 https://doi.org/10.1016/S0898-1221(04)90131-9
7. Q. M. Shao, Recent developments on self-normalized limit theorems. In: As- ymptotic methods in probability and statistics (editor B. Szyszkowicz) (1998), 467-480
8. W. F. Stout, Almost Sure Convergence, Academic Press, New York, 1974
9. B. Mandelbort, Limit theorems on the self-normalized range for weakly and strongly dependent processes, Z. Wahrsch. Verw. Gebiete 31 (1975), 271-285 https://doi.org/10.1007/BF00532867
10. P. S. Griffin and J. D. Kuelbs, Self-normalized laws of the iterated logarithm, Ann. Probab. 17 (1989), no. 4, 1571-1601 https://doi.org/10.1214/aop/1176991175
11. W. Feller, An Introduction to Probability and Its Applications. Vol. I, 3rd edn., Wiley & Sons, Inc., New York, 1968
12. M. Csorgo, Z. Y. Lin, and Q. M. Shao, Studentized increments of partial sums, Sci. China Ser. A 37 (1994), no. 3, 265-276
13. E. Gine, F. Gotze, and D. M. Mason, When is the Student t-statistic asymptoti- cally standard normal?, Ann. Probab. 25 (1997), no. 3, 1514-1531 https://doi.org/10.1214/aop/1024404523
14. Z. Y. Lin, A self-normalized Chung type law of the iterated logarithm, Theory Probab. Appl. 41 (1996), no. 4, 791-798