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Let $f(x,\sigma^2)$ be a representative element of a family of PDF's with full support over the reals that is indexed by their variance $\sigma^2$. Under what general conditions of the family of distributions can we guarantee that $f(x,\sigma^2)\rightarrow 0$ as $\sigma^2\rightarrow \infty$ for all $x$ in the domain?

I wonder if assuming that all elements in the family have finite variance and are absolutely continuous suffices to have the desired result.

I know that the normal satisfies this condition, but I was wondering if there was a more general family that would guarantee that the density of any point in the domain vanishes as the variance approaches infinity.

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  • $\begingroup$ Which type of functional convergence are you interested in? Do you mean zero or infinity? $\endgroup$ – Xi'an Feb 3 at 7:47
  • $\begingroup$ Thank you, I did mean infinity (just edited the question). I am interested in point-wise convergence. $\endgroup$ – Regio Feb 3 at 7:49
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    $\begingroup$ This question is awfully broad because you are asking for a nonlocal characterization of the family. That is, something has to happen for all appropriate sequences of functions $f_\sigma$ at every real number. It's easy to construct counterexamples just by taking any family for which your criterion holds and mixing in a fixed continuous distribution with all its members. If you have an application in mind, then, could you describe it specifically? $\endgroup$ – whuber Feb 3 at 14:34
  • $\begingroup$ Thank you for your comment. I tried to narrow the question by adding that I am interested in full support distributions over the reals. I also added my current hypothesis of what conditions are sufficient, but would love to see if you think such conditions are not enough and if you have some counterexamples. $\endgroup$ – Regio Feb 3 at 16:18
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    $\begingroup$ OK, but your reference to "full support on the reals" is a little confusing, because the exponential (aka $\Gamma(1)$) distribution is supported only on the non-negative reals. $\endgroup$ – whuber Feb 3 at 16:25

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