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• 1 Georgetown University Department of Philosophy Washington DC 20057 U.S.A. E-mail Washington DC 20057 U.S.A. E-mail
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## Abstract

Let X1,X2,... be a sequence of independent and identically distributed random variables, and put
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_n = X_1 + \cdot \cdot \cdot + X_n$$ \end{document}
. Under some conditions on the positive sequence τn and the positive increasing sequence an, we give necessary and sufficient conditions for the convergence of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\nolimits_{n = 1}^\infty {\tau _n } P\left( {\left| {S_n } \right| \geqslant \varepsilon an} \right)$$ \end{document}
for all & > 0, generalizing Baum and Katz's~(1965) generalization of the Hsu–Robbins–Erds (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where τn = n-1and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$an = \left( {n\log n} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}$$ \end{document}
for n ≤ 2, thereby answering a question of Spătaru. Moreover, some results for non-identically distributed independent random variables are obtained by a recent comparison inequality. Our basic method is to use a central limit theorem estimate of Nagaev (1965) combined with the Hoffman-Jrgensen inequality~(1974).

Manuscript Submission: HERE

• Impact Factor (2019): 0.693
• Scimago Journal Rank (2019): 0.412
• SJR Hirsch-Index (2019): 20
• SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
• Impact Factor (2018): 0.664
• Scimago Journal Rank (2018): 0.412
• SJR Hirsch-Index (2018): 19
• SJR Quartile Score (2018): Q2 Mathematics (miscellaneous)

Periodica Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1971
Volumes
per Year
2
Issues
per Year
4
Founder Bolyai János Matematikai Társulat - János Bolyai Mathematical Society
Founder's
H-1055 Budapest, Hungary Falk Miksa u. 12.I/4.
Springer Nature Switzerland AG
Publisher's
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
ISSN 0031-5303 (Print)
ISSN 1588-2829 (Online)

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