In Y.S. Chow and H. Robbins' On Sums of Independent Random Variables with Infinite Moments and “Fair” Games, on the second page (page 331 of the journal), the authors prove that the Petersburg random variable does not satisfy a generlization of the Strong Law of Large Numbers by the following argument (where $b_n:=n\log_2n$):
$$P(x>a)\ge\frac{1}{a}\textrm{ for }a\ge1.$$ Hence, for any constant $c>1$ and $n\ge2$, $$P(x>cb_n)\ge\frac{1}{cb_n}=\frac{1}{cn\log_2n}$$ and therefore, $$\sum_{n=1}^{\infty}{P(x>cb_n)=\infty}$$ which implies by the Borel-Cantelli lemma that $$P\left(\frac{x_n}{b_n}>c\textrm{ infinitely often}\right)=1.$$
But here, the events are not independent, so the converse of the Borel-Cantelli lemma does not apply. I assume they would then be using the counterpart in some way but I do not see how it applies here.
(The authors also use the Borel-Cantelli lemma in the same way throughout the rest of the paper)
Can someone explain how?
In fact, the events they are applying BC-converse to are independent. Perhaps the confusion is that they did not write the subscript $n$ on $x.$
The relevant events are $\{x_n >b_nc\}$ where $x_n$ is the winnings on the $n$-th game. The outcomes of the individual games are assumed independent of one another.
They presumably omitted the subscript $n$ when expressing $P(x_n> b_n c)$ because the winnings are also identically distributed.