  We prove a non-central limit theorem for the symmetric weighted odd-power
variations of the fractional Brownian motion with Hurst parameter H< 1/2. As
applications, we study the asymptotic behavior of the trapezoidal weighted
odd-power variations of the fractional Brownian motion and the fractional
Brownian motion in Brownian time Z_t:= X_{Y_t}, t >= 0, where X is a fractional
Brownian motion and Y is an independent Brownian motion.
