# KL divergence and convolution of distributions

Let $$P,Q,R$$ be probability measures on the real line. If these are discrete, we can show $$$$D_{\mathrm{KL}}(P\ast R\,\Vert\,Q\ast R)\le D_{\mathrm{KL}}(P\,\Vert\,Q)$$$$ by using the log sum inequality, where $$D_{\mathrm{KL}}(P\,\Vert\,Q)=\int \log\left(\frac{dP}{dQ}\right) dP$$ is KL divergence and $$\ast$$ is convolution. Is the above inequality true when $$P,Q,R$$ are not necessarily discrete?

• I am specially interested in a case where R is normal distribution. – selami Feb 12 '19 at 7:05

The KL divergence cannot increase after passing both distributions through the same Markov kernel (in your case, convolution with $$R$$). This is an immediate consequence of the data processing inequality: https://en.wikipedia.org/wiki/Data_processing_inequality