  In this paper we study the asymptotic behaviour of weighted random sums when
the sum process converges stably in law to a Brownian motion and the weight
process has continuous trajectories, more regular than that of a Brownian
motion. We show that these sums converge in law to the integral of the weight
process with respect to the Brownian motion when the distance between
observations goes to zero. The result is obtained with the help of fractional
calculus showing the power of this technique. This study, though interesting by
itself, is motivated by an error found in the proof of Theorem 4 in Corcuera,
J.M. Nualart, D., Woerner, J. H. C. (2006). Power variation of some integral
fractional processes, Bernoulli 12(4) 713-735.
