From uniform laws of large numbers to uniform ergodic theorems (Q2718881)

The lecture note covers the topic indicated in the title. It aims to provide a detailed introduction to uniform convergence in the ergodic theorem, uniform over classes of functions.NEWLINENEWLINENEWLINEThe author considers three methods to derive theorems of this type, all of them treated separately in Chapters 2, 3 and 4. The content can be sketched briefly as follows:NEWLINENEWLINENEWLINEThe approach in Chapter 2 is by using the theory of bracketing and the theorem of Blum and DeHardt (including \textit{J. Hoffmann-Jørgensen}'s extension in [Lect. Notes Math. 1153, 258-272 (1985; Zbl 0587.60009)]). One of the main results is the uniform ergodic theorem of the author and \textit{M. Weber} [Proc. Convergence in Ergodic Theory and Probability, de Gruyter, Berlin, Ohio State Univ. Math. Res. Inst. Publ. 5, 305-332 (1996; Zbl 0858.28010)].NEWLINENEWLINENEWLINEThe third chapter deals with Vapnik-Chernovenkis' approach to uniform estimates for partial sums. The analysis relies on Sudakov's minorization, which is as well covered as the necessary parts of the theory. A uniform law of large numbers (under absolute regularity) is proven in section 3.7 which is due to the author and \textit{J. E. Yukich} [Proc. Prob. Banach Spaces, Birkhäuser, Boston, Prog. Probab. 35, 105-128 (1994; Zbl 0828.28006)].NEWLINENEWLINENEWLINEThe last chapter on spectral representation covers a somewhat different topic. It deals with stationary processes in the wide sense and their uniform convergence in \(L_2\). The main results require some type of integrability of the supremum of the corresponding spectral measure. Besides the results mentioned, the reader will find many other useful results and connections to other people's work, which cannot be mentioned here.NEWLINENEWLINENEWLINEThe style of the book is rather technical and involves heavy notations, because it is the intention of the author to introduce the three methods explained above. There are almost no applications of the theorems mentioned, and there are only rare references to other work on uniform ergodic theorems. As a monograph on uniform ergodic theorems it is a valuable source.