... unknown. The purpose of this article is to study and establish the ASCLT for self-normalized partial sums of random variablesin the domain of attraction of the normal law, we will show that the ASCLT ... the domain of attraction of the normal law. Therefore, the class of randomvariables in Theorems 1.1 is of very broad range.Remark 1.5. Essentially, the open problem should be whether Theorem 1.1 ... on the almost sure limit theorem for self-normalizedpartial sums of random variables in the domain of attraction of the normal lawQunying Wu1,21College of Science, Guilin University of Technology,Guilin...