Weak convergence to a modified fractional Brownian motion

This article discusses the weak convergence of a particular version of a fractional process (appropriately normalized) to a modified fractional Brownian motion which has not been previously reported in the literature. Additionally, we illustrate a specific case in which this type of fractional Brownian motion characterizes the limiting behaviour of processes which might arise in practice. This is related to the situation termed by Hualde (2006) as unbalanced cointegration, where the integration orders of the observables are different, but their corresponding balanced versions (where one of the series is filtered adequately so it has identical integration order to the other one) are cointegrated in the usual sense.