The Borel-Cantelli Lemma

This monograph provides an extensive treatment of the theory and applications of the celebrated Borel-Cantelli Lemma. Starting from some of the basic facts of the axiomatic probability theory, it embodies the classical versions of these lemma, together with the well known as well as the most recent extensions of them due to Barndorff-Nielsen, Balakrishnan and Stepanov, Erdos and Renyi, Kochen and Stone, Petrov and the present author. The versions of the second Borel-Cantelli Lemma for pair wise negative quadrant dependent sequences, weakly *-mixing sequences, mixing sequences (due to Renyi) and for many other dependent sequences are all included. The special feature of the book is a detailed discussion of a strengthened form of the second Borel-Cantelli Lemma and the conditional form of the Borel-Cantelli Lemmas due to Levy, Chen and Serfling. All these results are well illustrated by means of many interesting examples. All the proofs are rigorous, complete and lucid. An extensive list of research papers, some of which are forthcoming, is provided. The book can be used for a self study and as an invaluable research reference on the present topic.

Tapas Kumar Chandra is a Professor of Statistics at the Indian Statistical Institute, Kolkata, India, since 1991. He is the author of the text books A First Course in Asymptotic Theory of Statistics and A First Course in Probability Theory. Among the problems solved by him, mention must be made of the age-old problems of finding valid Edgeworth expansions of the Likelihood Ratio Statistic and other perturbed Chi-square variables under very general conditions, and of comparing Bahadur-optimal test statistics, settling the conjecture of C. R. Rao on the optimality of score test for local alternatives and obtaining very general extensions of the Kolmogorov-Etemadi SLLN, the Marcinkiewicz-Zygmund SLLN, the well known results on the L1 and Lp convergence of the normalized sample sum and an inequality of Kolmogorov. His work is co-authored with J. K. Ghosh, R. R. Bahadur, S. Ghosal, R. Mukherjee, and others.