Let \( \{X, X_{n};~n \ge 1 \}\) 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 \(\sum _{n=1}^{\infty } f_{n}\left( \left\| S_{n} \right\| \right) \) where \(S_{n} = X_{1} + \cdots + X_{n}, ~n \ge 1\) and \(\left\{ f_{n}(\cdot ); n \ge 1 \right\} \) is a sequence of nonnegative nondecreasing functions defined on \([0, \infty )\). More specifically, it is shown that (i) the above random series diverges almost surely if \(\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) = \infty \) for some \(c > 0\) and (ii) the above random series converges almost surely if \(\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) < \infty \) for some \(c > 0\) provided additional conditions are imposed involving X, the sequences \(\left\{ S_{n};~n \ge 1 \right\} \) and \(\left\{ f_{n}(\cdot ); n \ge 1 \right\} \), and c. A special case of this criterion is a divergence/convergence criterion for the random series \(\sum _{n=1}^{\infty } a_{n} \left\| S_{n} \right\| ^{q}\) based on the series \(\sum _{n=1}^{\infty } a_{n} n^{q/2}\) where \(\left\{ a_{n};~n \ge 1 \right\} \) is a sequence of nonnegative numbers and \(q > 0\).

令\（\ {X，X_ {n};〜n \ ge 1 \} \）是一系列独立且分布均匀的Banach空间值随机变量。本文致力于为形式为\（\ sum _ {n = 1} ^ {\ infty} f_ {n} \ left（\ left \ | S_ {n} \ right \ | \ right）\）其中\（S_ {n} = X_ {1} + \ cdots + X_ {n}，〜n \ ge 1 \）和\（\ left \ {f_ {n}（\ cdot） ; n \ ge 1 \ right \} \）是在\（[0，\ infty）\）上定义的一系列非负非递减函数。更具体地，示出了：（i）如果\（\ sum _ {n = 1} ^ {\ infty} f_ {n} \ left（cn ^ {1/2} \ right）几乎可以肯定地发散。= \ infty \）对于某些\（c> 0 \）和（ⅱ）上述的随机系列几乎确定收敛如果\（\总和_ {N = 1} ^ {\ infty} F_ {N} \左（CN ^ {1/2} \右）<\ infty \）为某些\（c> 0 \）会提供附加条件，涉及X，序列\（\ left \ {S_ {n};〜n \ ge 1 \ right \} \）和\（\ left \ {f_ {n }（\ cdot）; n \ ge 1 \ right \} \）和c。此准则的一个特例是随机序列\（\ sum _ {n = 1} ^ {\ infty} a_ {n} \ left \ | S_ {n} \ right \ | ^ {q } \）基于\（\ sum _ {n = 1} ^ {\ infty} a_ {n} n ^ {q / 2} \）系列，其中\（\ left \ {a_ {n};〜n \ ge 1 \ right \} \）是一个非负数和\（q> 0 \）的序列。