  Parton distribution functions (PDFs) at large $x$ are poorly constrained by
high-energy experimental data, but extremely important for probing physics
beyond standard model at colliders. We study the calculation of PDFs at
large-$x$ through large-momentum $P^z$ expansion of the lattice quasi PDFs.
Similar to deep-inelastic scattering, there are two distinct perturbative
scales in the threshold limit where the matching coefficient can be factorized
into a space-like jet function at scale $P^z|1-y|$ and a pair of heavy-light
Sudakov form factors at scale $P^z$. The matching formula allows us to derive a
full renormalization group resummation of large threshold logarithms, and the
result is consistent with the known calculation to the next-to-next to leading
order (NNLO). This paves the way for direct large-$x$ PDFs calculations in
lattice QCD. As by-products, we find that the space-like jet function is
related to a time-like version calculated previously through analytic
continuation, and the heavy-light Sudakov form factor, calculated here to NNLO,
is a universal object appearing as well in the large momentum expansion of
quasi transverse-momentum-dependent PDFs and quasi wave-function amplitudes.
