Statistical nearly universal Glivenko-Cantelli classes

Suppose given a parametric family of densities f (x, theta). Let X-1, ..., X-n,... be i.i.d. observations with a law P not necessarily in the family. If the maximum likelihood estimates of theta exist and converge to some theta(0) = theta(0)(P), then 00 is called the pseudo-true value of the parameter. Analogues of pseudo-true values, here called M-limits, also can occur as limits of M-estimators more general than maximum likelihood estimators, where the sample average of a function rho(x, theta) is (approximately) minimized with respect to theta. In this paper, the question is for what rho do M-limits exist for all probability measures P on the sample space X, or for all P outside some "small" exceptional class?