Vector-valued Choquet-Deny theorem, renewal equation and self-similar measures

The Choquet-Deny theorem and Deny's theorem are extended to the vector-valued case. They are applied to give. a simple nonprobabilistic proof of the vector-valued renewal theorem, which is used to study the L(p)-dimension, the L(p)-density and the Fourier transformation of vector-valued self-similar measures. The results questions raised by Strichartz.