The Boundary Of Convergence Of Genaral Symmetric Random Analytic Series

We improve an important lemma-Cartan-Thullen lemma to the genaral symmettic random analytic series.We can prove a perfect conclusion. That is more that the boundary of convergence of genaral symmetric random analytic series is a natural boundary. So the domain of convergence is the holomorphic domain.Especially,for random Rademacher series,Steinhaus series,Dirichlet se-ries, the the domain of convergence is the holomorphic domain.