# What family of full support probability distributions satisfy that the density of any point in the domain vanishes as the variance goes to infinity?

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.

• Which type of functional convergence are you interested in? Do you mean zero or infinity? – Xi'an Feb 3 at 7:47
• Thank you, I did mean infinity (just edited the question). I am interested in point-wise convergence. – Regio Feb 3 at 7:49
• 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? – whuber Feb 3 at 14:34
• 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. – Regio Feb 3 at 16:18
• 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. – whuber Feb 3 at 16:25