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Suppose $X_i$ are a sequence of random variables with continuous distributions (possibly not absolutely continuous), such that

$$X_i \xrightarrow{d} X \sim \mathrm{Gamma}(k,\theta),$$

where $d$ denotes convergence in distribution. Does necessarily

$$\mathbb{E}[X_i^m] \xrightarrow{n \to \infty} \mathbb{E}[X^m],$$

where $m \in \mathbb{N}$?

This question is a continuation of an identical question without the continuity assumption; that question was answered negatively with a counterexample.

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  • $\begingroup$ You may "fix" the discontinuity by adding a small normally distributed random variable which goes to zero as $n\to\infty$. $\endgroup$ – Sergei Golovan Feb 17 '17 at 20:32
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There was nothing essential about the discreteness of the second random variable $2n$ added in the mixture in the answer to your previous question. They could have used any sequence of random variables whose means increased proportional to $n,$ including continuous ones that would produce a sequence of continuous distributions.

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  • $\begingroup$ Right, we may replace $2n$ with $Yn$, where $Y$ has continuous distribution and $\mathbb{E}[Y] \neq 0$. $\endgroup$ – kaba Feb 17 '17 at 20:44
  • $\begingroup$ @Kaba yep, that's right $\endgroup$ – spaceisdarkgreen Feb 17 '17 at 20:45

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