Reference to iterated logarithm law and Smirnov law of empirical CDF 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


*

*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 $x$.

*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.   
 A: I think the proofs are given in the book http://www.amazon.com/Probability-The-Classical-Limit-Theorems/dp/110762827X (Probability: The Classical Limit Theorems, by Henry McKean). Concerning the Smirnov Law, see also http://projecteuclid.org/euclid.aoms/1177730243 (On the Kolmogorov-Smirnov Limit Theorems for Empirical Distributions, by W. Feller).
A: 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].
