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Asymptotics of weighted random sums

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.