If $\mathbb{E}[e^{tX}]$ is bounded, then $\mathbb{E}[e^{2tX}]$ is bounded?

Question

Do we have the following statement,

If $\mathbb{E}[e^{tX}]$ is bounded, then $\mathbb{E}[e^{2tX}]$ is bounded?

I need to use this as an intermediate step in one of my proof, but I don't know how to show this. I was considering using change of variable, but that doesn't seem to work.

Any help would be appreciated!

Answer 1

Given a random variable $X$, the function $\mathbb E[e^{tX}]$, wherever it exists, is called the moment generating function, MGF for short.

You can show that the MGF exists only in an open interval around $0$ (which may be the whole real line also). This gives the MGF what we call the radius of convergence, which would be half the length of this open interval. Outside this ball, the MGF will fail to converge.

For example, for $X \sim Exp(\lambda)$, the radius of convergence is $\lambda$ itself. Here, taking $t = 0.75 \lambda$, for example, would mean that the MGF would exist at $t$ but not at $2t$.