Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X_n)_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider

A) $\int x \; d\mu(x) = \infty$

B) There almost surely exist infinitely many $n$ such that $$ \mathbf X_{n + 1} > \kappa \sum_{i = 1}^n \mathbf X_i. $$

Then (B) implies (A) (this follows from the law of large numbers together with the Borel-Cantelli lemma.) My question is: does (A) imply (B)?

According to the generalized Borel-Cantelli lemma, condition (B) is equivalent to

B$'$) The series $$ \sum_{n = 1}^\infty \mu\left(\left(\kappa\sum_{i = 1}^n \mathbf X_i,\infty\right)\right) $$ diverges almost surely. This fact was used to show that (B) holds when $d\mu(x) \sim x^{-(1 + \varepsilon)} dx$ ($\varepsilon\in (0,1)$.) I can also use it to show that (B) holds when $d\mu(x) \sim 1/(x^2\log(x))$, $d\mu(x) \sim 1/(x^2\log(x)\log\log(x))$, etc., using complicated calculations. This is highly suggestive that the result holds in general...