Strong Limit Theorems for Maximal Spacings from a General Univariate Distribution

Abstract

Let $X_1, X_2, \cdots$ be an i.i.d. sequence of random variables with a continuous density $f$. We consider in this paper the strong limiting behavior as $n \rightarrow \infty$ of the $k$th largest spacing $M^{(n)}_k$ induced by $X_1, \cdots, X_n$ in the sample range. In the case where $f$ is bounded away from zero inside a bounded interval and vanishes outside, we characterize the limiting behaviour of $M^{(n)}_k$ in terms of the local behavior of $f$ in the neighborhood of the point where it reaches its minimum. In the case where the support of $f$ is an unbounded interval, we prove that for any $k \geq 1, M^{(n)}_k \rightarrow 0$ a.s. as $n \rightarrow \infty$ if and only if the distribution of $X_1$ has strongly stable extremes.