On the Convergence Rate in “The Exact Asymptotics” for Random Variables with a Stable Distribution

In the note, we present the conditions under which relations of the type hold, where a random variable Y α has a stable distribution, C is a constant, and nonnegative functions r , f and g satisfy certain conditions. The obtained results allow to make more precise and to complement results, related to the convergence rate in the so called “exact asymptotics.”