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Academic Press, 2017. - 189 p. - ISBN 012809818XInequalities play a fundamental role in mathematics and its many applications. Especially in the statistics and probability literature, there are many more limit theorems than inequalities. However, usually at the **heart** of a good limit theorem is at least one good inequality. This should become clear if one recalls the deﬁnition of the limit and the fact that a neighborhood of a point in a speciﬁc topology is usually deﬁned in terms of inequalities. A limit theorem can be very illuminating. However, it only describes the behavior of a function near a given point (possibly at inﬁnity), whereas a corresponding inequality would cover an entire range, oftentimes in many or inﬁnitely many dimensions. Also, the nature of limit theorems is more qualitative, whereas that of inequalities is more quantitative. For example, a central limit theorem would state that a certain distribution is close to normality; such a statement by itself is qualitative, as it does not specify the degree of closeness under speciﬁc conditions. In contrast, a corresponding Berry–Esseen-type inequality can provide quantitative speciﬁcs. This is why good inequalities are important. A good inequality would be, not only broadly enough applicable, but also precise enough; ideally, it would be a solution to an extremal problem. Indeed, such results can be used most effectively in the theory and with a greater degree of conﬁdence and precision in real-world applications. Such an understanding of the role of good and, in particular, best possible bounds goes back at least to Chebyshev. In particular, the theory of Tchebycheff systems was developed to provide optimal solutions to a broad class of such problems. These ideas were further developed by a large number of authors, including Bernstein, Bennett, and Hoeffding. Quoting Bennett (J. Am. Stat. Assoc., 1962). The contributors to this book are leading experts in the area of inequalities and extremal problems in probability, statistics, and mathematical analysis. It is hoped that the material presented here will promote broader understanding of the importance of inequalities and extremal problems, and that it will stimulate further progress in this area.

The ﬁrst two chapters of the book, written by Osekowski and devoted to problems arising in the theory of semimartingales, have a strong analytical component.**Chapter 1** reviews the so-called method of moments, a powerful and general technique developed in the sixties in the works of Kemperman. The approach, based on dynamic-programming arguments and backward induction, allows the reduction of the study of quite general estimates to the construction of an appropriate functional sequence. This reduction is a common point in many related areas, for example, optimal stochastic control, optimal control theory, and Bellman function method. As an illustration of the method, several new sharp maximal bounds for martingale difference sequences, square function estimates, and prophet inequalities for square-integrable martingales are presented. **Chapter 2** contains a study of a new class of optimal stopping problems for Brownian motion and its maximal function. The classical Markovian approach enables solving such problems of the “integral” form, yielding, in particular, the Doob and Hardy–Littlewood inequalities. The novel method presented in Chapter 2 enables the investigation of optimal stopping problems of “non-integral” type, including Lorentz-norm estimates valid for arbitrary stopping times. **Chapter 3**, written by Shevtsova, presents the latest and so far the best known universal constants in the so-called nonuniform Berry–Esseen (BE) bounds for sums of independent random variables. Such bounds work better than their uniform counterparts in the tail zones, which are especially important in statistical testing. To a large extent, the method is based on the idea going back to Nagaev and further back to Cramér—to employ the exponential tilt transform to reduce the problem of the nonuniform BE bounds to that of the uniform ones, which latter can be tackled by using, for example, appropriate smoothing inequalities, including ones due to Esseen or Prawitz. However, this exponential-tilt reduction appears to have inherent limitations, discussed in **Chapter 4**. Two alternative methods are suggested there, based on novel smoothing inequalities, from which nonuniform BE bounds can be obtained directly, without using the mentioned nonuniform-to-uniform reduction. The purpose of **Chapter 5** is to establish new uniform BE bounds for the Student statistic or, equivalently, for the so-called self-normalized sums, with explicit constant factors. The concluding **Chapter 6** was written by de la Peña and Ibragimov. Sharp probability and moment inequalities for random polynomials, generalized sample cross-moments, and their self-normalized and Studentized versions, in random variables with an arbitrary dependence are discussed there. The results are based on sharp extensions of probability and moment inequalities for sums of independent random variables to the case of the above statistics in independent symmetric variables. The case of statistics in dependent variables is treated through the use of measures of dependence. The results presented in Chapter 6 are applicable in a number of settings in statistics, econometrics, and time series analysis, including tests for independence and problems of detecting nonlinear dependence.**Iosif Pinelis** is Professor in the Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan, USA.

**Victor de la Peña** is Professor at the Department of Statistics, Columbia University, USA.

**Rustam Ibragimov** is Professor of Finance and Econometrics at Imperial College Business School, UK.

**Adam Osekowski** is Associate Professor at the Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Poland.

**Irina Shevtsova** is Associate Professor at Moscow State University, and Senior Researcher at the Institute of Informatics Problems of the Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, Russia.

The ﬁrst two chapters of the book, written by Osekowski and devoted to problems arising in the theory of semimartingales, have a strong analytical component.

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