Lin Zhengyan, Wang Hanchao. Weak Convergence and its Applications

World Scientific, 2014. — 185 p.The book is devoted to the theory of weak convergence of probability measures on metric spaces.There are many books concerning the weak convergence, for example, Ethier and Kurtz (1986), van der Vaart and Wellner (1996), Billingsley (1999), Jacod and Shiryaev (2003) and so on. The emphasis of these books are different. Statistics and econometrics have made great progress in last two decades. It became necessary to study the distributions or the asymptotic distributions of some complex statistics, so the weak convergence of stochastic processes based on these complex statistics would be more important. Some models may need new technique in weak convergence to study. For example,
In the study of the asymptotic behavior of non-stationary time series, the limiting process (resp. limiting distribution) usually associates with the processes with conditional independent increment (resp. mixture normal distributions).
The asymptotic errors caused by discretizations of stochastic differential equations (e.g. analysis of hedging error) converge to stochastic integral in distribution.
In the finance risk theory, the returns of assets usually obey heavy-tailed distributions, the statistics based on such data do not have the asymptotic normality.
The processes based on these statistics usually converge to the non-Gaussian stable process weakly.
Some statistics in the goodness-of-fit testing are empirical processes or functional index empirical processes. It is quite difficult to state the weak convergence of such processes.Our main aim is to give a systematic exposition of the theory of weak convergence, covering wide range of weak convergence problems including new developments. In this book, we summarize the development of weak convergence in each of the following aspects: Donsker type invariance principles, convergence of point processes, weak convergence to semimartingale and convergence of empirical processes. Compared to the well-known monographs mentioned earlier, we do not cover all details, but our book contains the core content of many branches and new development as much as possible. This is perhaps one of the highlights of our book.