Journal of the
Korean Mathematical Society JKMS

Let $\{X, X_n; n\ge 1\}$ be a sequence of $i.i.d.$ random variables which belong to the attraction of the normal law, and $X_n^{(1)}, \ldots, X_n^{(n)}$ be an arrangement of $X_1, \ldots, X_n$ in decreasing order of magnitude, i.e., $|X_n^{(1)}|\ge \cdots \ge |X_n^{(n)}|$. Suppose that $\{r_n\}$ is a sequence of constants satisfying some mild conditions and $d^{'}(t_{n_k})$ 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, $i.i.d.$ random variables

MSC numbers: Primary 60F15; Secondary 60G50