Asymptotic behavior of weighted quadratic variation of tempered fractional Brownian motion. (English) Zbl 1537.60047
Summary: This paper investigates the convergence in \(L^2\) of renormalized weighted quadratic variation associated to tempered fractional Brownian motion with Hurst index \(0 < H < 1/4\) and \(\lambda > 0\). We first give four lemmas about tempered fractional Brownian motion by means of Malliavin calculus. Then the convergence for tempered fractional Brownian motion is derived in \(L^2\) through these lemmas. Our main result extends findings of
I. Nourdin [Ann. Probab. 36, No. 6, 2159–2175 (2008; Zbl 1155.60010)], concerning the weighted power variations of fractional Brownian motion.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 60F05 | Central limit and other weak theorems |
| 60H05 | Stochastic integrals |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60G15 | Gaussian processes |
Citations:
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