A Nonstandard Law of the Iterated Logarithm for Trimmed Sums

Abstract

Let $X_i, i \geq 1$, be independent random variables with a common distribution in the domain of attraction of a strictly stable law, and for each $n \geq 1$ let $X_{1, n} \leq \cdots \leq X_{n, n}$ denote the order statistics of $X_1, \ldots, X_n$. In 1986, S. Csorgo, Horvath and Mason showed that for each sequence $k_n, n \geq 1$, of nonnegative integers with $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, the trimmed sums $S_n(k_n) = X_{k_n + 1, n} + \cdots + X_{n - k_n, n}$ converge in distribution to the standard normal distribution, when properly centered and normalized, despite the fact that the entire sums $X_1 + \cdots + X_n$ have a strictly stable limit, when properly centered and normalized. The asymptotic almost sure behavior of $S_n(k_n)$ strongly depends on the rate at which $k_n$ converges to $\infty$. The sequences $k_n \sim c \log \log n$ as $n \rightarrow \infty$ for $0 < c < \infty$ constitute a borderline case between a classical law of the iterated logarithm and a radically different behavior. This borderline case is investigated in detail for nonnegative summands $X_i$.