Very simple question: Suppose $X$ is a subgaussian RV with a density $p_X(x)$. Does this imply any nontrivial upper bounds on $p_X(x)$ for large $x$?
Some context: It is not hard to show that bounds like $p_X(x)\le e^{-x^2}$ imply subgaussianity. I am wondering about the converse, or can the density of a subgaussian RV be arbitrarily bad?
Note: The example of a bounded, constant RV shows that such upper bounds can only hold for sufficiently large $x$.