From convergence in distribution to convergence in probability

Posted: December 21, 2018 in Cross-Validated relay, Educational Material, stats.stackexchange relay
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When $\hat \theta \to_d D(\theta, v) \implies \hat\theta \to_p \theta$ ?

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).