## The Annals of Probability

- Ann. Probab.
- Volume 12, Number 4 (1984), 1181-1193.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 12, Number 4 (1984), 1181-1193.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176993147

**Digital Object Identifier**

doi:10.1214/aop/1176993147

**Mathematical Reviews number (MathSciNet)**

MR757775

**Zentralblatt MATH identifier**

0558.62018

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

**Keywords**

Laws of the iterated logarithm order statistics spacings strong laws almost sure convergence empirical processes quantile processes

#### Citation

Deheuvels, Paul. Strong Limit Theorems for Maximal Spacings from a General Univariate Distribution. Ann. Probab. 12 (1984), no. 4, 1181--1193. doi:10.1214/aop/1176993147. https://projecteuclid.org/euclid.aop/1176993147