Minimal spacings of non-uniform densities
Typical of the results of this paper is the following: Suppose that \(X_ 1,X_ 2,..\). are i.i.d. random variables with a continuous probability density f(x), \(A<x<B<\infty\), such that \(\overline{\lim}_{x\to A+}f(x)<\infty\), \(\lim_{x\to B-}f(x)=\infty\) and f is non-decreasing in (B-\(\epsilon\),B) for some \(\epsilon >0\). If \(X_{1,n},...,X_{n,n}\) are the order statistics of \(X_ 1,...,X_ n\), the only possible non- trivial asymptotic distribution for the suitably scaled difference \(X_{n,n}-B\) is a Weibull distribution. Following ideas of \textit{P. Deheuvels} [Ann. Probab. 14, 194-208 (1986; Zbl 0594.60029)] the author proves that if the parameter \(\alpha\) of this Weibull distribution is in (0,\()\), then, as \(n\to \infty\), the k-th smallest of the spacings \(X_{i+1,n}-X_{i,n}\), \(1\leq i\leq n\), converges in distribution for each k to the k th smallest of the random variables \[ (\sum^{i+1}_{j=1}w_ j)^{1/\alpha}- (\sum^{i}_{j=1}w_ j)^{1/\alpha},\quad i\geq 1, \] where the \(w_ j's\) are i.i.d. exp(1). This and other similar results are applied to coverage problems of the circle and the line by random arcs or segments, and to the asymptotics of the scan statistics used in goodness-of-fit.
scientific article; zbMATH DE number 4030592
scientific article