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Let $X_1,\ldots, X_n$ be i.i.d. observations from a family of pdf or pmf $\{f_{\theta}: \theta \in \Theta \}$. We know that there are sufficient regularity conditions on the family $\{f_{\theta}: \theta \in \Theta \}$ that ensures the maximum likelihood estimators of $\theta$ converge in probability or almost surely to $\theta$. One can find such conditions in large sample theory books (e.g., R. Serfling, Ferguson, Lehmann, etc). Are there sufficient regularity conditions on the family $\{f_{\theta}: \theta \in \Theta \}$ such that: (1) the moments of the maximum likelihood estimator of $\theta$ exist? (2) the maximum likelihood estimator of $\theta$ converges in mean (L1 convergence) to $\theta$?

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