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

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

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

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