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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].

MSC:

60G17 Sample path properties
60B10 Convergence of probability measures
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