doi: 10.1685/journal.caim.486

Asymptotics of weighted random sums

José Manuel Corcuera, David Nualart Nualart, Mark Podolskij


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 observation distances 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: J. M. Corcuera, D. Nualart, and J. H. C. Woerner, Power variation of some integral fractional processes, Bernoulli, vol. 12, no. 4, pp. 713–735, 2006.

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Communications in Applied and Industrial Mathematics
ISSN: 2038-0909