  It is well-known that the Fourier-Galerkin spectral method has been a popular
approach for the numerical approximation of the deterministic Boltzmann
equation with spectral accuracy rigorously proved. In this paper, we will show
that such a spectral convergence of the Fourier-Galerkin spectral method also
holds for the Boltzmann equation with uncertainties arising from both collision
kernel and initial condition. Our proof is based on newly-established spaces
and norms that are carefully designed and take the velocity variable and random
variables with their high regularities into account altogether. For future
studies, this theoretical result will provide a solid foundation for further
showing the convergence of the full-discretized system where both the velocity
and random variables are discretized simultaneously.
