  The recently proposed large momentum effective theory (LaMET) of Ji has led
to a burst of activity among lattice practitioners to perform and control the
first pioneering calculations of the quasi-PDFs of the nucleon. These
calculations represent approximations to the standard PDFs defined as
correlation functions of fields with lightlike separation, being instead
correlations along a longitudinal direction of the operator $\gamma^z$; as
such, they differ from standard PDFs by power-suppressed $1 \big/ p^2_z$
corrections, becoming exact in the limit $p_z \to \infty$. Investigating the
systematics of this behavior thus becomes crucial to understanding the validity
of LaMET calculations. While this has been done using models for the nucleon,
an analogous dedicated study has not been carried out for the $\pi$ and $\rho$
quark distribution functions. Using a constituent quark model, a systematic
calculation is performed to estimate the size and $x$ dependence of the
finite-$p_z$ effects in these quasi-PDFs, finding them to be potentially tamer
for lighter mesons than for the collinear quasi-PDFs of the nucleon.
