  We consider the statistical properties of the gravitational field F in an
infinite one-dimensional homogeneous Poisson distribution of particles, using
an exponential cut-off of the pair interaction to control and study the
divergences which arise. Deriving an exact analytic expression for the
probability density function (PDF) P(F), we show that it is badly defined in
the limit in which the well known Holtzmark distribution is obtained in the
analogous three-dimensional case. A well defined P(F) may, however, be obtained
in the infinite range limit by an appropriate renormalization of the coupling
strength, giving a Gaussian form. Calculating the spatial correlation
properties we show that this latter procedure has a trivial physical meaning.
Finally we calculate the PDF and correlation properties of differences of
forces (at separate spatial points), which are well defined without any
renormalization. We explain that the convergence of these quantities is in fact
sufficient to allow a physically meaningful infinite system limit to be defined
for the clustering dynamics from Poissonian initial conditions.
