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.
"Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4." Ann. Probab. 37 (6) 2200 - 2230, November 2009. https://doi.org/10.1214/09-AOP473