  The study of the magnification bias produced on high-redshift sub-millimetre
galaxies by foreground galaxies through the analysis of the cross-correlation
function was recently demonstrated as an interesting independent alternative to
the weak-lensing shear as a cosmological probe. In the case of the proposed
observable, most of the cosmological constraints mainly depend on the largest
angular separation measurements. Therefore, we aim to study and correct the
main large-scale biases that affect foreground and background galaxy samples to
produce a robust estimation of the cross-correlation function. Then we analyse
the corrected signal to derive updated cosmological constraintsWe measured the
large-scale, bias-corrected cross-correlation functions using a background
sample of H-ATLAS galaxies with photometric redshifts > 1.2 and two different
foreground samples (GAMA galaxies with spectroscopic redshifts or SDSS galaxies
with photometric ones, both in the range 0.2 < z < 0.8). These measurements are
modelled using the traditional halo model description that depends on both halo
occupation distribution and cosmological parameters. We then estimated these
parameters by performing a Markov chain Monte Carlo under multiple scenarios to
study the performance of this observable and how to improve its results. After
the large-scale bias corrections, we obtain only minor improvements with
respect to the previous magnification bias results, mainly confirming their
conclusions: a lower bound on $\Omega_m > 0.22$ at $95\%$ C.L. and an upper
bound $\sigma_8 < 0.97$ at $95\%$ C.L. (results from the $z_{spec}$ sample).
However, by combining both foreground samples into a simplified tomographic
analysis, we were able to obtain interesting constraints on the
$\Omega_m$-$\sigma_8$ plane as follows: $\Omega_m= 0.50_{- 0.20}^{+ 0.14}$ and
$\sigma_8= 0.75_{- 0.10}^{+ 0.07}$ at 68\% CL.
