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

Let X 1 , ···, X n be i.i.d. random variables defined in ℝ d having common continuous density f ( x ), and let R ij be the rank of X j 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.