In the paper “On the order of magnitude of the difference between consecutive prime numbers” by Harald Cramér there is the following statement:

Suppose $\{X_n\}_{n=2}^\infty$ is a sequence of independent random variables, such that $X_n \sim Bern(\frac{1}{\ln(n)})$. Then $\lim_{n \to \infty} \sup |\frac{\sqrt{\ln(n)}(\Sigma_{i=2}^n X_i - li(n))}{\sqrt{2n \ln(\ln(n))}}| = 1$

However, he does not prove this result there, but rather states, that it is proved in his paper “Prime numbers and probability” (which I could not find)

My question is:

How can this statement be proved?

Probably, it has something to do with the Law of Iterated Logarithm, but I do not know for sure ...