Relative entropy (KL divergence) of sum of random variables Suppose we have two independent random variables, $X$ and $Y$, with different probability distributions.
What is the relative entropy between pdf of $X$ and $X+Y$, i.e.
$$D(P_X||P_{X+Y})$$
assume all support conditions are met.
I know in general pdf of $X+Y$ is convolution of pdf of $X$ and $Y$, but is there 
an easier way to calculate the relative entropy or at least simplify it?

Here's an attempt.
$$D(P_X||P_{X+Y}) = \mathbb{E}[\log \frac{P_X(X)}{P_{X+Y}(X+Y)}] = \mathbb{E}[\log \frac{P_X(X)P_Y(Y)}{P_{X+Y}(X+Y)P_Y(Y)}] \\
= \mathbb{E}[\log \frac{P_X(X)}{P_Y(Y)}] + \mathbb{E}[\log \frac{P_Y(Y)}{P_{X+Y}(X+Y)}] = \infty$$
because $g$ is uniform and has $g(x)=0$ for some $x$ such that $f(x)>0$.
