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**Approximation and weak convergence methods for random processes, with applications to stochastic systems theory.**
*(English)*
Zbl 0551.60056

The MIT Press Series in Signal Processing, Optimization, and Control, 6. Cambridge, Massachusetts, - London: The MIT Press. XVII, 269 p. $ 54.00 (1984).

Recently the weak convergence methods attract the attention of many researchers in the field of stochastics and its various applications. However it is not so easy to write a book which is to satisfy the taste and level of the theorists, and to be understandable and useful for the applied scientists. The present work successfully covers both these aspects: good theory with clear schemes and ideas for applications.

In chapter 1 the author gives a short review of stochastic processes and especially of martingales, Wiener process, stochastic integrals and equations. The so-called martingale problem, playing an important role in the sequel, is discussed too. The concept of weak convergence is considered in chapters 2 and 3. Here we find a nice description of general weak convergence theorems as well as their versions in some specific functional spaces. In many cases the proofs are omitted, but the author indicates the references and, which is important, he considers concrete examples in order to show how the weak convergence methods could be applied in problems of control and communications theory. The perturbed test function method is developed in chapter 4. Here we find several results concerning weak convergence and tightness, with corresponding proofs. The author pays particular attention to some discrete parameter models where the limit process is a diffusion process. Several weak convergence type results can be obtained by using the direct-averaging method. This is done with many details in chapter 5. Here the author considers models with discrete and continuous time parameter and noise process of various sort.

Chapter 6 deals with the asymptotic behavior of different perturbed systems. Stability results for Ito stochastic differential equations by Lyapunov functions are given. Moreover, conditions are found to guarantee the existence of a unique invariant measure. Actually, much of the book is concerned with singularly perturbed dynamic systems. They are analyzed in chapter 7. The author illustrates the stability and tightness techniques by suitable examples. The next three chapters, 8, 9 and 10, treat numerous applications in control and communications theory. Another kind of problems is discussed in chapter 11. Here the author considers systems where the noise effects are ”small” and the time of interest is large. He presents several results establishing Chernoff, Ventsel- Freidlin or Freidlin-GĂ¤rtner bounds for the large deviations. This chapter also completes with several useful examples. The book ends with a reference list (about 100 items) and an index of terms.

Finally, let me note that the author presents known techniques and also develops some powerful new methods. The presentation is careful and precise. There is no doubt that this book will be met with an interest by both categories of readers - professional stochasticians and scientists who wish to apply approximation methods for analysis of various complex dynamic systems.

In chapter 1 the author gives a short review of stochastic processes and especially of martingales, Wiener process, stochastic integrals and equations. The so-called martingale problem, playing an important role in the sequel, is discussed too. The concept of weak convergence is considered in chapters 2 and 3. Here we find a nice description of general weak convergence theorems as well as their versions in some specific functional spaces. In many cases the proofs are omitted, but the author indicates the references and, which is important, he considers concrete examples in order to show how the weak convergence methods could be applied in problems of control and communications theory. The perturbed test function method is developed in chapter 4. Here we find several results concerning weak convergence and tightness, with corresponding proofs. The author pays particular attention to some discrete parameter models where the limit process is a diffusion process. Several weak convergence type results can be obtained by using the direct-averaging method. This is done with many details in chapter 5. Here the author considers models with discrete and continuous time parameter and noise process of various sort.

Chapter 6 deals with the asymptotic behavior of different perturbed systems. Stability results for Ito stochastic differential equations by Lyapunov functions are given. Moreover, conditions are found to guarantee the existence of a unique invariant measure. Actually, much of the book is concerned with singularly perturbed dynamic systems. They are analyzed in chapter 7. The author illustrates the stability and tightness techniques by suitable examples. The next three chapters, 8, 9 and 10, treat numerous applications in control and communications theory. Another kind of problems is discussed in chapter 11. Here the author considers systems where the noise effects are ”small” and the time of interest is large. He presents several results establishing Chernoff, Ventsel- Freidlin or Freidlin-GĂ¤rtner bounds for the large deviations. This chapter also completes with several useful examples. The book ends with a reference list (about 100 items) and an index of terms.

Finally, let me note that the author presents known techniques and also develops some powerful new methods. The presentation is careful and precise. There is no doubt that this book will be met with an interest by both categories of readers - professional stochasticians and scientists who wish to apply approximation methods for analysis of various complex dynamic systems.

Reviewer: J.M.Stoyanov

### MSC:

60Hxx | Stochastic analysis |

60Gxx | Stochastic processes |

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

60F10 | Large deviations |

60F05 | Central limit and other weak theorems |

93E15 | Stochastic stability in control theory |

60J60 | Diffusion processes |