  In this two--part study, we present the development and analysis of a
stochastic theory for characterizing the relative positions of monodisperse,
low-inertia particle pairs that are settling rapidly in homogeneous isotropic
turbulence. In the limits of small Stokes number and Froude number such that
$Fr \ll St_\eta \ll 1$, closures are developed for the drift and diffusion
fluxes in the probability density function (PDF) equation for the pair relative
positions. The theory focuses on the relative motion of particle pairs in the
dissipation regime of turbulence, i.e., for pair separations smaller than the
Kolmogorov length scale. In this regime, the theory approximates the fluid
velocity field in a reference frame following the primary particle as locally
linear.
