I am reading V. Vapnik's "Statistical Learning Theory". The author layouts following two statistical laws related to empirical CDF. I am looking for reference about proofs on these two laws. 


Let $F\_l(x)$ be the empirical CDF from $l$ i.i.d samples drawing from same distribution with CDF $F(x)$. 


Iterated Logarithm Law:
$$
\mathbb{P}\left( \limsup\_{l \to \infty} \sup\_x \sqrt{\frac{l}{\ln\ln l}}|F\_l(x) - F(x)| = 1 \right) = 1
$$


Smirnov Law:
$$
\lim\_{l \to \infty} \mathbb{P}\left( l \int (F\_l(x) - F(x))^2 dF(x) < \epsilon \right) = 1 - \frac{2}{\pi} \sum\_{k=1}^\infty \int\_{(2k-1)\pi}^{2k\pi} \frac{\exp(-\lambda^2\epsilon/2)}{\sqrt{-\lambda \sin \lambda}} d\lambda 
$$


*some comments*


1. Iterated Logarithm Law: I think it can't trivially be implied by classical iterated logarithm law for i.i.d. sequence, because the law holds true uniformly for all $x$.
2. Smirnov Law: I know the law was proved first by Smirnov in Russian article. I am looking for English reference with formal proof of this law.



Reference to iterated logarithm law and Smirnov law of empirical CDF
For Smirnov's result I think the easiest approach (at least on a "hand waving" level) is via empirical processes:


As long as $F$ is continuous then it suffices to consider uniform distributions.


Let $\Delta\_n(t) = n^{1/2}(F\_n(t) - t)$.


Then the Smirnov Cramer von Mises statistic you wrote is
$$
\int\_0^1 \Delta\_n(t)^2 dt.
$$


But $\Delta\_n(t)$ converges weakly to a Brownian bridge [Kolmogorov, Doob, Donsker, Andersson, .... proofs dripping from many books].


So, sweeping away some boring technicalities,
$$
\int\_0^1 \Delta\_n(t)^2 dt \to \int\_0^1 BB(t)^2 dt
$$  

where $BB$ is a standard Brownian bridge.


The distribution of the RHS is that which Smirnov derived. 


[I think the link <https://projecteuclid.org/download/pdf_1/euclid.aop/1020107767> I posted contains some refs on how to evaluate the distribution of the integral of a squared BB].


