Strong Shannon-McMillan-Breiman's theorem for locally compact groups

We prove that for a vast class of random walks on a compactly generated group, the exponential growth of convolutions of a probability density function along almost every sample path is bounded by the growth of the group. As an application, we show that the almost sure and L-1 convergences of the Shannon-McMillan-Breiman theorem hold for compactly supported random walks on compactly generated groups with subexponential growth.