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

摘要

In the note, we present the conditions under which relations of the typedocumentclass12pt{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$ underset{varepsilon searrow 0}{lim}left(sum_{nge 1}r(n)textbf{P}left({Y}_{alpha}ge fleft(varepsilon g(n)right)right)-nu left(varepsilon right)right)=C $$end{document}hold, where a random variable Yα 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.”