When ?

We know that in general, convergence in distribution does not imply convergence in probability. **But for the case of most interest in econometrics, where we examine a sequence of estimators, convergence in distribution does imply convergence in probability to a constant, **under two regularity conditions that are also satisfied in most cases of interest.

**This post of mine in stats.stackexchange.com** has the proof. Essentially, under these regularity conditions we are able to prove the even stronger result that convergence in distribution implies convergence in quadratic mean to a constant (which in turn implies convergence in probability to that constant).