Statistical properties of the gravitational force through ordering statistics

We investigate the statistical distribution of Newtonian gravitational forces acting on a test particle embedded in an infinite, homogeneous, and uncorrelated random gas of point masses. Using order statistics, we derive the probability density functions of distances to the $n$-th nearest neighbors in arbitrary spatial dimensions and analyze their contributions to the total gravitational force. We show that, in three dimensions, the formally divergent variance of the Holtsmark probability density function arises entirely from the first nearest neighbor, while contributions from more distant neighbors remain finite. Our results provide a clear decomposition of local versus distant contributions to the gravitational forces exerted on a test particle, and explore the dominant role of nearest neighbors in the divergence of the Holtsmark distribution's variance.