you need to show that the sandwich estimator converges in probability to the asymptotic variance of the MLE under the misspecified model.

A start on this proof would be to write your (starting and target) convergence formulae in terms of power=series expansions involving the sample-size. This need only be up to the first-order error terms, using little-oh-in-probability notation.

as you are using log of f i suppose its with likelihood principle. but i want to correct that Variance is HIH^-1 using spectral theory.good day