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Consider i.i.d. samples $X_1, \dots, X_n$ from a distribution $\mathbb{P}_{\theta}$ in the exponential family, where $\theta$ denotes the distribution parameters. Let $\widehat{\theta}$ denote the MLE estimate. Then, we know that $\widehat{\theta} \stackrel{a.s.}{\rightarrow} \theta$.

Now suppose we instead have independent samples $X_1, \dots, X_n$ such that $X_n \stackrel{d}{\rightarrow} \mathbb{P}_{\theta}$ as $n \rightarrow \infty$.

Do we still have $\widehat{\theta} \stackrel{a.s.}{\rightarrow} \theta$? Or even $\widehat{\theta} \stackrel{p}{\rightarrow} \theta$?

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    I'm guessing you are defining $\hat \theta$ through the functional form of the MLE as if $X_i$ were iid from $P_\theta$? – 2017-01-10
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    yes that is right – 2017-01-10

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