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Asymptotic behavior of weighted quadratic variations of fractional Brownian motion: The critical case H=1/4

We derive the asymptotic behavior of weighted quadratic variations of fractional Brow- nian 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.