  We study statistical anisotropies generated in the observed two-point
function of the cosmic microwave background (CMB) fluctuations if the
primordial statistics are non-Gaussian. Focusing on the dipole modulations of
the anisotropies, we find that the hemispherical power asymmetry observed in
the CMB temperature fluctuations can be modeled by a local-type trispectrum
with amplitude $\tau_{\rm NL}(k_p=0.05~{\rm Mpc}^{-1}) \approx 2 \times 10^4$
and a large red tilt $n\approx -0.68$. We numerically evaluate the non-Gaussian
covariance of the modulation estimators for both temperature and E-mode
polarization fluctuations and discuss the prospects of constraining the model
using Planck satellite data. We then discuss other effects of the
scale-dependent trispectrum that could be used to distinguish this scenario
from other explanations of the power asymmetry: higher-order modulations of the
two-point function and the non-Gaussian angular power spectrum covariance. As
an important consequence of the non-Gaussian power spectrum covariance, we
discuss how the CMB-inferred spectral index of primordial scalar fluctuations
can be significantly biased in the presence of a scale-dependent local-type
trispectrum.
