A random variable $X$ is said to be sub-gaussian with mean $\mu$ and *pseudo-variance* $\sigma^2$ iff
$$\mathbb \log(E[\exp(t(X-\mu))]) \le \frac{t^2}{2\sigma^2},\;\forall t \in \mathbb R.
$$

It's a standard computation that the KL divergence between two random variables with means $\mu_1$ and $\mu_2$ and same variance $\sigma^2 > 0$ is $\dfrac{(\mu_1 - \mu_2)^2}{2\sigma^2}$.

# Question

What is good upper-bound for the KL divergence between two sub-gaussian variables with means $\mu_1$ and $\mu_2$ and same pseudo-variance $\sigma^2 > 0$, and **full support** ?

data-processing inequalities, $\operatorname{KL}(P\| Q) \ge \operatorname{KL}(P * R \| Q * R)$, and so the convolution trick you mention won't help get an upper bound on $\operatorname{KL}(P\| Q)$. $\endgroup$ – dohmatob Jul 31 '18 at 20:10in proportion to $P$'s tail. Sub-Gaussianity of the two variables only says that their two tails are dominated by some Gaussian's tail, and says nothing about the relations of the tails to each other which is what you need. $\endgroup$ – enthdegree Jul 31 '18 at 20:37