  In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018],
spectral convergence and long-time decay of the numerical solution towards the
global equilibrium of the stochastic Galerkin approximation for the Boltzmann
equation with random inputs in the initial data and collision kernel for hard
potentials and Maxwellian molecules under Grad's angular cutoff were
established using the hypocoercive properties of the collisional kinetic model.
One assumption for the random perturbation of the collision kernel is that the
perturbation is in the order of the Knudsen number, which can be very small in
the fluid dynamical regime. In this article, we remove this smallness
assumption, and establish the same results but now for random perturbations of
the collision kernel that can be of order one. The new analysis relies on the
establishment of a spectral gap for the numerical collision operator.
