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


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.


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


  1. V. Bentkus and F. Gotze, The Berry-Esseen bound for Student's statistic, Ann. Probab. 24 (1996), no. 1, 491-503
  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
  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
  4. M. Hahn and J. Kuelbs, Universal asymptotic normality for conditionally trimmed sums, Stat. Prob. Lett. 7 (1988), no. 1, 9-15
  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
  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
  10. P. S. Griffin and J. D. Kuelbs, Self-normalized laws of the iterated logarithm, Ann. Probab. 17 (1989), no. 4, 1571-1601
  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
  14. Z. Y. Lin, A self-normalized Chung type law of the iterated logarithm, Theory Probab. Appl. 41 (1996), no. 4, 791-798