For an arbitrary point of a homogeneous Poisson point process in a
d-dimensional Euclidean space, consider the number of Poisson points
that have that given point as their rth nearest neighbor $(r = 1, 2,
\dots)$. It is shown that as d tends to infinity, these nearest neighbor
counts $(r = 1, 2, \dots)$ are iid asymptotically Poisson with mean 1. The
proof relies on Rényi's characterization of Poisson processes and a
representation in the limit of each nearest neighbor count as a sum of
countably many dependent Bernoulli random variables.