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Random series and stochastic integrals: single and multiple. (English) Zbl 0751.60035
Probability and Its Applications. Boston: Birkhäuser. xvi, 360 p. (1992).
The roots of the subject of the book are in classical probability theory, and can be traced back to Kolmogorov’s summation of independent random variables and Wiener’s works on polynomial chaoses. The book contains classical results as well as more recent, even previously unpublished results, for sums of independent random vectors, martingales, semimartingales, multiple random series, and multiple stochastic integrals, both real and Banach space-valued. The Gaussian and non- Gaussian context is treated in detail.
The authors have chosen to focus on robust techniques and results, pertinent to an arbitrary Banach space, rather than pursuing topics requiring intrinsic geometric properties of the space. The approach is based on a coercive use of various type of inequalities, and use of characteristic functions (Fourier transforms) is almost reduced to null. That makes the book accessible for a larger audience yet without a sacrifice of the generality. The capacity limitations did not allow the authors to touch each and every angle of the theory, mainly because the area of random summation and integration is still in the stage of intensive development, and the count of connections, interdisciplinary excursions, applications, etc. is endless. To name but a few connections, one can mention the Itô and Malliavin’s calculus, which in turn yields to stochastic differential equations, quantum field theory, von Mises’ statistics, statistics (\(U\)- and other statistics, quadratic forms), etc.
A bridge between the main stream of lecture and paths of further study is formed by extensive (25 % of the text) Complements and Comments sections, appearing at the end of each chapter. Numerous references, historical background, remarks on applications, discussion on connections, etc., result in the bibliography of more than 350 items.
The book is divided into a part devoted to a random series, and a part dealing with stochastic integrals. Chapter 1 presents a number of fundamental inequalities for random series of independent random variable in Banach space. Chapter 2 deals with convergence of such series, addressing relations between various modes of convergence (a.s., in \(L^ p\), in probability) and types of compactness. Chapter 3 introduces a variety of means of comparison of random variables, and of series of such, with special attention paid to Bernoulli and Gaussian series. After bringing up classical and modern results on martingales (the latter include decoupling and tangency) in Chapter 4, the authors apply in Chapter 5 refined methods, developed in Chapter 3, for martingales. The discrete part of the book is concluded by Chapter 6, dealing with contraction principle, domination, decoupling, and convergence of random multilinear forms in independent random variables. Again, the Gaussian case is given a special consideration.
The second part begins with constructions and examples of general stochastic integrals (Chapter 7). In Chapter 8 the authors elaborate the case of deterministic integrands and integrators that are processes with independent increments. Chapter 9 is devoted to the semimartingale integration. Chapter 10 covers foundations of multiple integration, in particular, constructions, characterizations of integrable functions, and multiple series expansions of Wiener integrals.
Two appendices provide an additional material, closely related to the contents of both parts (Appendix A: Unconditional and Bounded Multilinear Convergence of Random Series; Appendix B: Vector Measures).
Reviewer: J.Szulga (Auburn)

60Gxx Stochastic processes
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60H05 Stochastic integrals
60G50 Sums of independent random variables; random walks
60G57 Random measures
60G15 Gaussian processes
60G44 Martingales with continuous parameter