Bound for density of random variable with finite second moment

Question

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!

Answer 1

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.