Limit theorems for multi-indexed sums of random variables.

*(English)*Zbl 1318.60005
Probability Theory and Stochastic Modelling 71. Berlin: Springer (ISBN 978-3-662-44387-3/hbk; 978-3-662-44388-0/ebook). xviii, 483 p. (2014).

This book deals with limit theorems for sums of multi-indexed (\(d\)-indexed) independent random variables. The main difference between the classical case of \(d=1\) and \(d>1\) considered in the book is that the space of indices is not completely ordered in the latter case. This circumstance has little importance for weak limit theorems but makes a huge difference for almost sure convergence. Even the definition of convergence can be understood in several ways for \(d>1\). Two types of convergence are considered in the book, namely max- and min-convergence. The first of them is considered as the maximum of coordinates of an index tends to infinity with the obvious change for min-convergence.

In Chapter 1, the author explains the difference between these two cases, compares the classical conditions and those suitable for the case of multi-indexed sums, mentions several results for \(d=1\) that have no analogs for \(d>1\) and vice versa. Chapter 2 contains maximal inequalities for multi-indexed sums. In particular, a very general and useful approach to the so-called Hajek-Renyi inequality is described. Chapters 3 and 4 discuss weak convergence. Chapter 5 is devoted to the almost sure convergence of infinite multi-indexed series. In contrast to the classical case, a four series theorem provides necessary and sufficient conditions for almost sure convergence (three series are not sufficient in this case!). In Chapters 6 and 7, the almost sure boundedness and the rate of convergence of multi-indexed series are considered. These questions are rather rare in the monograph literature and the author presents several new results even in the case of \(d=1\). The strong law of large numbers is discussed in Chapters 8 and 9. A new method of proof is presented and many examples are given. The method is elementary and very powerful at the same time. The law of the iterated logarithm for multi-indexed sums is studied in Chapter 10. Chapter 11 discusses an interesting topic on renewal processes constructed from random walks with multidimensional time. The author points out a relationship between a renewal theorem in this case and the Riemann hypothesis on zeros of the zeta function. The probabilistic setting of the so-called dominated ergodic theorem is treated in Chapter 12. The problem considered here is to determine the moment conditions for the existence of the supremum of multi-indexed sums of independent identically distributed random variables. The final Chapter 13 is devoted to complete convergence, where again some interesting links to number theory appear.

Presenting the first unified treatment of limit theorems for multiple sums of independent random variables, this volume fills an important gap in the field. Several new results are introduced, even in the classical setting, as well as some new approaches that are simpler than those already established in the literature.

The book is well written and mathematically rigorous. The author has been working with sums of multi-indexed independent random variables for more than 30 years. He is certainly the greatest specialist in this field worldwide. He has collected a large variety of results and tried to parallel the theory for single index sums and he does this rather successfully.

To date there is no book like the present one. All of the important results on multiple sums are scattered throughout the literature. A unified treatment such as this one is a most valuable addition to the literature. It may become the standard reference for researchers working on the topic of multiple sums.

In summary, this is a useful book for a researcher in probability theory and mathematical statistics. It is very carefully written and collects results which are not easy to find in the literature or which had been even forgotten.

In Chapter 1, the author explains the difference between these two cases, compares the classical conditions and those suitable for the case of multi-indexed sums, mentions several results for \(d=1\) that have no analogs for \(d>1\) and vice versa. Chapter 2 contains maximal inequalities for multi-indexed sums. In particular, a very general and useful approach to the so-called Hajek-Renyi inequality is described. Chapters 3 and 4 discuss weak convergence. Chapter 5 is devoted to the almost sure convergence of infinite multi-indexed series. In contrast to the classical case, a four series theorem provides necessary and sufficient conditions for almost sure convergence (three series are not sufficient in this case!). In Chapters 6 and 7, the almost sure boundedness and the rate of convergence of multi-indexed series are considered. These questions are rather rare in the monograph literature and the author presents several new results even in the case of \(d=1\). The strong law of large numbers is discussed in Chapters 8 and 9. A new method of proof is presented and many examples are given. The method is elementary and very powerful at the same time. The law of the iterated logarithm for multi-indexed sums is studied in Chapter 10. Chapter 11 discusses an interesting topic on renewal processes constructed from random walks with multidimensional time. The author points out a relationship between a renewal theorem in this case and the Riemann hypothesis on zeros of the zeta function. The probabilistic setting of the so-called dominated ergodic theorem is treated in Chapter 12. The problem considered here is to determine the moment conditions for the existence of the supremum of multi-indexed sums of independent identically distributed random variables. The final Chapter 13 is devoted to complete convergence, where again some interesting links to number theory appear.

Presenting the first unified treatment of limit theorems for multiple sums of independent random variables, this volume fills an important gap in the field. Several new results are introduced, even in the classical setting, as well as some new approaches that are simpler than those already established in the literature.

The book is well written and mathematically rigorous. The author has been working with sums of multi-indexed independent random variables for more than 30 years. He is certainly the greatest specialist in this field worldwide. He has collected a large variety of results and tried to parallel the theory for single index sums and he does this rather successfully.

To date there is no book like the present one. All of the important results on multiple sums are scattered throughout the literature. A unified treatment such as this one is a most valuable addition to the literature. It may become the standard reference for researchers working on the topic of multiple sums.

In summary, this is a useful book for a researcher in probability theory and mathematical statistics. It is very carefully written and collects results which are not easy to find in the literature or which had been even forgotten.

Reviewer: Nikolai N. Leonenko (Cardiff)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G60 | Random fields |

60G50 | Sums of independent random variables; random walks |

60F15 | Strong limit theorems |

60F05 | Central limit and other weak theorems |

60E15 | Inequalities; stochastic orderings |

60G15 | Gaussian processes |

60K05 | Renewal theory |