# Bound for density of random variable with finite second moment

Let $$\mathbf{X}$$ be a vector-valued random variable with finite second moment and density $$\rho$$. Assume that $$\rho$$ is bounded and continuous. As $$\mathbf{X}$$ has finite second moment, I hope to find a bound of its density $$\rho(\mathbf{x})$$ for large $$\lvert\mathbf{x}\rvert$$ by a density of Gaussian type. My question is, more specifically, if it is possible to find a covariance matrix $$\Gamma>0$$ and a constant $$M$$ such that for large $$\lvert\mathbf{x}\rvert$$, it holds that $$\rho(\mathbf{x}) \leq M\pi_{\Gamma}(\mathbf{x}),$$ where $$\pi_{\Gamma}$$ denotes a normal density with a suitable mean and covariance matrix $$\Gamma$$. Counterexamples or oppositions are, of course, also welcome.

Thank you!

## 1 Answer

Why do you think this should hold ? In the one-dimensional case, consider $$\rho$$ the density of an exponential distribution $$\mathcal E(1)$$. Since for any $$a$$, $$\exp(-(x-a)^2/2) = o_{+\infty}(\exp(-x))$$ it's not possible to find the $$M$$ you're looking for.

• Well, sometimes you are too much biased since you hope for something :P. All the mentioned cases make sense and are, kind of, obvious. Thank you! – MTP May 22 '19 at 6:24