On the Influence of the Extremes of an I.I.D. Sequence on the Maximal Spacings

Abstract

Let $X_1, X_2,\cdots$ be an i.i.d. sequence of random variables with a continuous density $f$, positive on $(A, B)$, and null otherwise. Under the assumption that $Y_n = \min\{X_1,\cdots, X_n\}$ and $Z_n = \max\{X_1,\cdots, X_n\}$ belong to the domain of attraction of extreme value distributions and that $f(x) \rightarrow 0$ as $x \rightarrow A$ or $x \rightarrow B$, we show that the weak limiting behavior of $Y_n$ and $Z_n$ characterizes completely the weak limiting behavior of the maximal spacing generated by $X_1,\cdots, X_n$ and obtain the corresponding limiting distributions. We study as examples the cases of the normal, Cauchy, and gamma distributions.