Abstract: This paper obtains some equivelent conditions for a type of convolution closure of local subexponential distributions on 0,\infty), which are also valid for distributions on (-\infty,\infty) under certain conditions. On the basis of these results, the local asymptotics for the distribution of symmetrization are given. The results above include the corresponding, non-local results of Embrechts and Goldie (1980)\ucite1 and Geluk (2004)\ucite2. Some of our proofs are more simple than those of Geluk (2004)\ucite2.