Let {X,Xn;n≥1}\documentclass[12pt]{minimal}
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\begin{document}$$ \{X, X_{n};~n \ge 1 \}$$\end{document} be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to providing a divergence criterion for a class of random series of the form ∑n=1∞fnSn\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{n=1}^{\infty } f_{n}\left( \left\| S_{n} \right\| \right) $$\end{document} where Sn=X1+⋯+Xn,n≥1\documentclass[12pt]{minimal}
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\begin{document}$$S_{n} = X_{1} + \cdots + X_{n}, ~n \ge 1$$\end{document} and fn(·);n≥1\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ f_{n}(\cdot ); n \ge 1 \right\} $$\end{document} is a sequence of nonnegative nondecreasing functions defined on [0,∞)\documentclass[12pt]{minimal}
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\begin{document}$$[0, \infty )$$\end{document}. More specifically, it is shown that (i) the above random series diverges almost surely if ∑n=1∞fncn1/2=∞\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) = \infty $$\end{document} for some c>0\documentclass[12pt]{minimal}
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\begin{document}$$c > 0$$\end{document} and (ii) the above random series converges almost surely if ∑n=1∞fncn1/2<∞\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) < \infty $$\end{document} for some c>0\documentclass[12pt]{minimal}
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\begin{document}$$c > 0$$\end{document} provided additional conditions are imposed involving X, the sequences Sn;n≥1\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ S_{n};~n \ge 1 \right\} $$\end{document} and fn(·);n≥1\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ f_{n}(\cdot ); n \ge 1 \right\} $$\end{document}, and c. A special case of this criterion is a divergence/convergence criterion for the random series ∑n=1∞anSnq\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{n=1}^{\infty } a_{n} \left\| S_{n} \right\| ^{q}$$\end{document} based on the series ∑n=1∞annq/2\documentclass[12pt]{minimal}
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\begin{document}$$\sum _{n=1}^{\infty } a_{n} n^{q/2}$$\end{document} where an;n≥1\documentclass[12pt]{minimal}
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\begin{document}$$\left\{ a_{n};~n \ge 1 \right\} $$\end{document} is a sequence of nonnegative numbers and q>0\documentclass[12pt]{minimal}
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\begin{document}$$q > 0$$\end{document}.
