  In this article we use a covariance function that arises from limit of
fluctuations of the rescaled occupation time process of a branching particle
system, to introduce a family of weighted long-range dependence Gaussian
processes. In particular, we consider two subfamilies for which we show that
the process is not a semimartingale, that the processes exhibit long-range
dependence and have long-range memory of logarithmic order. Finally, we
illustrate that this family of processes is useful for modeling real world
data.
