# 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? Feb 3, 2021 at 7:47
• Thank you, I did mean infinity (just edited the question). I am interested in point-wise convergence. Feb 3, 2021 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, 2021 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. Feb 3, 2021 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, 2021 at 16:25