Are there any analogues of the Golden-Thompson Inequality for moment generating functions? The Golden-Thompson Inequality asserts the following: If $A$ and $B$ are two $n \times n$ Hermitian matrices, then $\text{tr}(e^{A+B}) \leq \text{tr}(e^{A}e^{B})$ with no commutativity hypotheses on $A$ and $B$.

If $A$ and $B$ commute, then $e^{A+B} = e^{A}e^{B}$. If not, then $e^{A+B} = e^{A}e^{B}e^{-\frac{1}{2}[A,B]} \dots$

So the trace operator gets rid of the "messiness." In a similar way, suppose we are given $X_1$ and $X_2$ as independent random variables. Then $E[e^{t(X_1+X_2)}] = E[e^{tX_1}] E[e^{tX_2}]$. Can we obtain a similar inequality to the Golden-Thompson with no hypotheses on $X_1$ and $X_2$?