Summary. We consider the asymptotic behaviour of the distance R_n(t) between a point t(d)R^dand the k_nth nearest neighbour of t among a sample X₁,…, X_nof i.i.d. random vectors, X_i R^d, for n(d)(d). It is shown, that under appropriate conditions on the underlying distribution of X_iR_n(t) is asymptotically equivalent to a nonrandom sequence a_n(t). This equivalence relation holds uniformly in t, with probability one and with a certain rate of convergence. The result is useful for the study of nonparametric density and regression estimates of nearest neighbour type.