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?

Let $f(t)$ be the PDF of $X$ and $g(t)$ be the PDF of $Y$.
$$D_{KL}(P_X\parallel P_{X+Y}) = \int_{-\infty}^{+\infty}f(x)\log\frac{f(x)}{(f*g)(x)}\,dx$$
does not admit any obvious simplification, but the term 
$$\log\frac{f(x)}{(f*g)(x)}=\log\frac{\int_{-\infty}^{+\infty} f(t)\,\delta(x-t)\,dt}{\int_{-\infty}^{+\infty} f(t)\,g(x-t)\,dt} $$
can be effectively controlled if some informations about the concentration/decay of $g(t)$ are known.
Is this the case?
