Lifshits, Mikhail; Weber, Michel Oscillations of Gaussian Stein’s elements. (English) Zbl 0906.60038 Eberlein, Ernst (ed.) et al., High dimensional probability. Proceedings of the conference, Oberwolfach, Germany, August 1996. Basel: Birkhäuser. Prog. Probab. 43, 249-261 (1998). Summary: We investigate the properties of a remarkable class of Gaussian sequences arising from E. M. Stein’s probabilistic approach [Ann. Math., II. Ser. 74, 140-170 (1961; Zbl 0103.08903)] to continuity principle in ergodic theory followed by spectacular applications in real analysis due to J. Bourgain [Isr. J. Math. 63, No. 1, 79-97 (1988; Zbl 0677.60042)] and further investigations of the second named author [Rend. Mat. Appl., VII. Ser. 15, No. 4, 569-605 (1995; Zbl 0849.60034) and in: Probability in Banach spaces, 9. Prog. Probab. 35, 129-151 (1994; Zbl 0808.28011)]. For such sequences, we obtain a nearly complete picture of the properties of the oscillations and describe the weak and strong convergence properties of associated sojourn times.For the entire collection see [Zbl 0883.00024]. Cited in 1 Document MSC: 60G17 Sample path properties 60B10 Convergence of probability measures Keywords:Gaussian sequences; ergodic theory; convergence properties; sojourn times Citations:Zbl 0103.08903; Zbl 0677.60042; Zbl 0849.60034; Zbl 0808.28011 PDFBibTeX XMLCite \textit{M. Lifshits} and \textit{M. Weber}, Prog. Probab. 43, 249--261 (1998; Zbl 0906.60038)