- Probability Theory & Measure Theory, 2nd Ed., by R. Ash and C. Dolèans-Dale
- Probability, Theory and Examples, 4th Ed., by R. Durrett.
- Probability and Measure, 3rd edition, by P. Billingsley.
- Asymptotic Statistics, by A.W. van der Vaart.
Some central limit theorem results in increasing dimension:
- Chernozhukov, V., Chetverikov, D. and Kato, K. (2016). Central
Limit Theorems and Bootstrap in High Dimensions, arxiv
- Bentkus, V. (2003). On the dependence of the Berry–Esseen bound on
dimension, Journal of Statistical Planning and Inference, 113,
- Portnoy, S. (1986). On the central limit theorem in R p when
$p \rightarrow \infty$, Probability Theory and Related Fields,
To learn more about standard proof of asympotoic optinmality in parametric models see:
- Lehman, E. L. and Casella, G. (2003) Theory of Point Estimation, Springer, 2nd edition. Chapter 6, Section 5
- van der Vaart, A.W. (1998) Asymptotic Statistics, Cambridge University Press. Chapet 5 and 7.
- Schervish, M. (1995). Theory of Statistics, Speinger. Chapter 7.
- Ferguson, T.S. (1996). A Course in Large Sample Theory, Chapman & Hall. Chapters 17 and 18.
To learn more a bout the concentration of measure phenomenon, useful references are:
- An Elementary Introduction to Modern Convex Geometry, by K. Ball. pdf
- Lectures on Discrete Geometry, by J. Matousek, 2002, Springer. Chapter 13 and 14.
- The Concentration of Measure Phenomenon, by M. Ledoux, 2005,
- Measure Concentration, Lecture Notes for Math 710, by
Alexander Barvinok, 2005. pdf
- A New Look at Independence -- Special
Invited Paper, by M.
Talagrand, the Annals of Applied
Probability, 24(1),1--34, 1996.
- Concentration Inequalities: A Nonasymptotic Theory of
Independencei, by S. Boucheron, G. Lugosi and P. Massart, Oxford
University Press, 2013.