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.

integralof the squared empirical process, whereas theKolmogorov-Smirnovstatistic uses themaximum absolute valueof the EP. $\endgroup$ – P.Windridge Aug 27 '15 at 20:29