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Hsu-Robbins and Spitzer’s theorems for the variations of fractional Brownian motion. (English) Zbl 1189.60085

Summary: Using recent results on the behavior of multiple Wiener-Itô integrals based on Stein’s method, we prove P. L. Hsu and H. Robbins’ [Proc. Natl. Acad. Sci. USA 33, 25–31 (1947; Zbl 0030.20101)] and F. Spitzer’s [Trans. Am. Math. Soc. 82, 323–339 (1956; Zbl 0071.13003)] theorems for sequences of correlated random variables related to the increments of the fractional Brownian motion.

MSC:

60G22 Fractional processes, including fractional Brownian motion
60H05 Stochastic integrals
60F05 Central limit and other weak theorems
60H07 Stochastic calculus of variations and the Malliavin calculus
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