Mutual and shared neighbor probabilities: finite- and infinite-dimensional results

Abstract

Let X 1, ···, Xn be i.i.d. random variables defined in ℝ d having common continuous density f(x), and let Rij be the rank of Xj in the ordered list of distances from X¡. Both the mutual neighbor probabilities p 1(r, s) = P(R 12 = r, R 21 = s) and the neighbor-sharing probabilities p 2(r, s) = P(R 13 = r, R 23 = s) are studied from an asymptotic viewpoint. Infinite-dimensional limits are found for both situations and take particularly simple forms. Both cases exhibit considerable stability across dimensions and thus are well approximated by their infinite-dimensional values. Tables are provided to support the results given.