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.


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


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