Some limit theorems for large deviations of sums of independent random variables with a common distribution function from the domain of normal attraction of a stable distribution

We examine certain questions, related to the convergence rate in the so-called “precise asymptotics”, when a limiting law is stable (including the normal one). In particular, the results obtained in Gut and Steinebach, (Ann. Univ. Sci. Budapest. Sect. Comput. 39:95–110, 2013) are generalized and refined.