Summary: Let E be a complete separable metric vector space and let q be a measurable seminorm on E. Suppose further that \(S=\sum X_ i\) is a series a.s. convergent with respect to q with independent components which are log concave and finite-dimensional. We prove that \(F(t)=P\{q(S)\leq t\}\) is positive and continuously differentiable for all \(t>0\).