• Probability: Theory and Examples, R. Durrett (2010), chapter 8.
• A course in probability theory, K.L. Chung (2000).
From a review of Chung's book:
There is a substantial chapter on invariance theorems, including Donsker's invariance principle, the Doob-Donsker proof of the Kolmogorov-Smirnov theorem and finally a general invariance principle of Skorokhod, the proof of which is only sketched.
• Convergence of Probability Measures, by P. Billingsley (1999)
From a review of Billingsley's book:
This book is about weak- convergence methods in metric spaces, with applications sufficient to show their power and utility. The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued.