  We study the statistical properties of magnification perturbations by
substructures in strong lensed systems using linear perturbation theory and an
analytical substructure model including tidal truncation and a continuous
substructure mass spectrum. We demonstrate that magnification perturbations are
dominated by perturbers found within a tidal radius of an image, and that
sizable magnification perturbations may arise from small, coherent
contributions from several substructures within the lens halo. We find that the
root-mean-square (rms) fluctuation of the magnification perturbation is 10% to
20% and both the average and rms perturbations are sensitive to the mass
spectrum and density profile of the perturbers. Interestingly, we find that
relative to a smooth model of the same mass, the average magnification in
clumpy models is lower (higher) than that in smooth models for positive
(negative) parity images. This is opposite from what is observed if one assumes
that the image magnification predicted by the best-fit smooth model of a lens
is a good proxy for what the observed magnification would have been if
substructures were absent. While it is possible for this discrepancy to be
resolved via nonlinear perturbers, we argue that a more likely explanation is
that the assumption that the best-fit lens model is a good proxy for the
magnification in the absence of substructure is not correct. We conclude that a
better theoretical understanding of the predicted statistical properties of
magnification perturbations by CDM substructure is needed in order to affirm
that CDM substructures have been unambiguously detected.
