Having access to those references, accumulating many results in one domain is always a bless, like Feller's book in probability, Dembo-Zeitouni's large deviation, Grimmett's percolation and recent Optimal Transport of Villani.

There are variants of asymptotic results in probability theory: law of large numbers, central limit theorem and laws of iterated logarithm. Each has its variants: weak LLN, strong LLN, i.i.d. variables, non i.i.d. variables, CLT for Markov chains etc. There are different ways of proving each one too.

Now I was curious to know about the references that provide most of these results and their different proofs.

I am aware of the following reference:

Anirban DasGupta, Asymptotic Theory of Statistics and Probability

Remark: If we can classify results of concentration inequalities as part of asymptotic results, then I am aware of Pascal Massart's Saint Flour lecture 2003 and some other references (Talagrand's notes for instance).

Any other references and discussions are appreciated.


2 Answers 2


There is a very recent book, October 2014 if I am not mistaken, by Oleg Klesov, titled "Limit Theorems for Multi-Indexed Sums of Random Variables". It has a fascinating content with good survey of many different limit problems.

Here is the table of content.

  • Some Remarks on the Theory of Limit Theorems for Multi-Indexed Sums
  • Maximal Inequalities for Multi-Indexed Sums of Independent Random
  • Variables Weak Convergence of Multi-Indexed Sums of Independent
  • Random Variables The Law of Large Numbers for Multi-Indexed Sums of
  • Independent Random Variables Almost Sure Convergence of Multi-Indexed
  • Series Boundedness of Multi-Indexed Series of Independent Random Variables
  • Rate of Convergence of Multi-Indexed Series
  • The Strong Law of Large Numbers for Independent Random Variables
  • The Strong Law of Large Numbers for Independent Identically Distributed Random Variables
  • The Law of the Iterated Logarithm
  • Renewal Theorems for Random Walks with Multi-Dimensional Time
  • Existence of Moments of Suprema of Multi-Indexed Sums and the Strong Law of Large Numbers
  • Complete Convergence

And the link;


A classical reference is Petrov's book Limit Theorems of Probability Theory, and find it here https://global.oup.com/academic/product/limit-theorems-of-probability-theory-9780198534990

"The exposition in the basic sections of the book is self-contained, with detailed proofs. Hence, the book is suitable for a course on limit theorems for graduate students. The book can also serve as a reference book for researchers in probability theory and theoretical statistics."-Mathematical Reviews


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.