Let \((E,\| \cdot \|)\) be a uniformly convex Banach space with modulus of convexity of power type p, F the distribution function of \(\sum^{\infty}_{i=1}\xi_ ix_ i\), where \((x_ i)\) is a sequence of linearly independent elements of E, and \((\xi_ i)\) are independent, real random variables with continuous densities such that \(\sum^{\infty}_{i=1}\xi_ ix_ i\) is an a.e. convergent in E series. The authors prove that F has a continuous and bounded derivative, and assuming additionally that the norm \(\| \cdot \|\) is k-times Fréchet differentiable and the densities of the one-dimensional components are smooth enough, then for any \(a\in E\), the distribution function \(F_ a(t)=u\{x; \| x+a\| 0\).