The convolution metric d g
J. E. Yukich · Mathematical Proceedings of the Cambridge Philosophical Society · 1985
Summary We introduce and study a new metric on denned by where is the space of probability measures on ℝ k and where g: ℝ k → is a probability density satisfying certain mild conditions. The metric d g , relatively easy to compute, is shown to have useful and interesting properties not enjoyed by some other metrics on . In particular, letting p n denote the nth empirical measure for P , it is shown that under appropriate conditions satisfies a compact law of the iterated logarithm, converges in probability to the supremum of a Gaussian process, and has a useful stochastic integral representation.