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

Title & Authors
A SELF-NORMALIZED LIL FOR CONDITIONALLY TRIMMED SUMS AND CONDITIONALLY CENSORED SUMS
Pang Tian Xiao; Lin Zheng Yan;

Abstract
Let {$\small{X,\;X_n;n\;{\geq}\;1}$} be a sequence of $\small{{\imath}.{\imath}.d.}$ random variables which belong to the attraction of the normal law, and $\small{X^{(1)}_n,...,X^{(n)}_n}$ be an arrangement of $\small{X_1,...,X_n}$ in decreasing order of magnitude, i.e., $\small{\|X^{(1)}_n\|{\geq}{\cdots}{\geq}\|X^{(n)}_n\|}$. Suppose that {$\small{{\gamma}_n}$} is a sequence of constants satisfying some mild conditions and d'($\small{t_{nk}}$) is an appropriate truncation level, where $\small{n_k=[{\beta}^k]\;and\;{\beta}}$ 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;$\small{{\imath}.{\imath}.d.}$ random variables;
Language
English
Cited by
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