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Register a new userAsymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case $H=1/4$
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We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion $B$ with Hurst index $H=1/4$. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to $B$.
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In this paper we state and prove a central limit theorem for the finite-dimensional laws of the quadratic variations process of certain fractional Brownian sheets. The main tool of this article is a method developed by Nourdin and Nualart based on the Malliavin calculus.
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Nils Tongring (1987) proved sufficient conditions for a compact set to contain $k$-tuple points of a Brownian motion. In this paper, we extend these findings to the fractional Brownian motion. Using the property of strong local nondeterminism, we show that if $B$ is a fractional Brownian motion in $mathbb{R}^d$ with Hurst index $H$ such that $Hd=1$, and $E$ is a fixed, nonempty compact set in $mathbb{R}^d$ with positive capacity with respect to the function $phi(s) = (log_+(1/s))^k$, then $E$ contains $k$-tuple points with positive probability. For the $Hd > 1$ case, the same result holds with the function replaced by $phi(s) = s^{-k(d-1/H)}$.
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