Journal of the
Korean Mathematical Society JKMS

Let $X, X_{1},X_{2},\ldots$ be i.i.d. random variables with zero means, variance one, and set $S_{n}=\sum_{i=1}^n X_i$, $n\geq1$. Gut and Sp\v{a}taru [3] established the precise asymptotics in the law of the iterated logarithm and Li, Nguyen and Rosalsky [7] generalized their result under minimal conditions. If ${\bf P}(|S_{n}|\geq \varepsilon\sqrt{2n\log\log n})$ is replaced by ${\bf E}\{|S_{n}|/\sqrt{n}-\varepsilon\sqrt{2\log\log n}\}_{+}$ in their results, the new one is called the moment version of precise asymptotics in the law of the iterated logarithm. We establish such a result for self-normalized sums, when $X$ belongs to the domain of attraction of the normal law.

Keywords: precise asymptotics, law of iterated logarithm, self-normalized sums

MSC numbers: 60F15, 62E20