Direct numerical simulations were performed using a smooth profile method to investigate the steady-state sedimentation of monodisperse spherical particles in an incompressible fluid at finite Peclet numbers 0 <= Pe <= 115. Hydrodynamic interactions caused strong fluctuations in the instantaneous velocity of the particles around the mean settling velocity. We found that the amplitude of these velocity fluctuations increases in direct proportion with the square of the Stokes velocity at higher Peclet numbers, where sedimentation is dominated by non-equilibrium hydrodynamic fluctuations. The diffusive behaviour of the particles was observed to be in a steady state over long time scales, and the steady-state self-diffusion coefficient was found to increase linearly with the Peclet number. Our results provide new insights into the anisotropy of vertical and horizontal diffusion. This anisotropy increases with an increasing Peclet number, plateauing at a high Peclet number.