  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 $B$.
