Convolutions of equicontractive self-similar measures on the line

Let μ be a self-similar measure on R generated by an equicontractive iterated function system. We prove that the Hausdorff dimension of μ*n tends to $1$ as n tends to infinity, where μ*n denotes the n-fold convolution of μ. Similar results hold for the Lq dimension and the entropy dimension of μ*n.