Limit Distribution of the Minimum Distance between Independent and Identically Distributed d-Dimensional Random Variables

Abstract

For a sequence X₁,X₂,... of i.i.d. d-dimensional random variables let Y_n denote the minimum Euclidean distance among the first n variables. It is shown that if the probability density function f of X_i belongs to L²(R^d) then the limit distribution of n²Y ^d_n is an exponential distribution whose intensity is proportional to the square of the L²-norm ‖f‖. The limit joint distribution of the smallest k distances and the midpoints of the k nearest pairs of variables is also obtained.