Limit theorems for the logarithm of sample spacings

Many statistical problems can be reformulated in terms of tests of uniformity. Some strong laws of large numbers and a central limit theorem for the logarithm of transformed spacings are obtained. These theorems provide a characterization of the uniform distribution. A general information-type inequality is deduced which gives a quantitative measurement (using the Kullback-Leibler number) of the discrepancy between an arbitrary distribution and the uniform distribution.