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 ...



A theorem by Harald Cramér?
This is a special case of a general law of the iterated logarithm for non-iid random variables (r.v.'s), which states the following: 



> 
> Suppose that $Y\_1,Y\_2,\dots$ are independent zero-mean r.v.'s, $S\_n:=\sum\_1^n Y\_i$, $B\_n:=Var\, S\_n\to\infty$, $|Y\_n|\le M\_n\in(0,\infty)$, and $M\_n=o((B\_n/\ln\ln B\_n)^{1/2})$. Then 
>  $$\limsup\_n\frac{S\_n}{\sqrt{2B\_n\ln\ln B\_n}}=1
> $$
>  almost surely. 
> 
> 
> 


See e.g. [V. Petrov, Ch. X, Theorem 1](https://www.springer.com/gp/book/9783642658112). This theorem is due to [Kolmogorov (1929)](https://gdz.sub.uni-goettingen.de/id/PPN235181684_0101?tify=%7B%22pages%22:[130],%22panX%22:0.466,%22panY%22:0.748,%22view%22:%22toc%22,%22zoom%22:0.795%7D). 


(Just in case, here is a reference to Cramér's paper: 


Cramér, H. 1935 Prime numbers and probability. Skand. Mat.-Kongr. 8, 107--115. 


I found it in [Granville's paper](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=2ahUKEwiN_bebl-rjAhVSbs0KHYTUD0oQFjAAegQIARAC&url=https%3A%2F%2Fwww.dartmouth.edu%2F~chance%2Fchance_news%2Ffor_chance_news%2FRiemann%2Fcramer.pdf&usg=AOvVaw2HM2mbW4_kBU9e8zEyaQLq).)


