Approximations of fractional Brownian motion using Poisson processes whose parameter
sets have the same dimensions as the approximated processes have been studied in the
literature. In this paper, a special approximation to the one-parameter fractional Brownian
motion is constructed using a two-parameter Poisson process. The proof involves the
tightness and identification of finite-dimensional distributions.

This thesis is a study in approximation of the fractional Brownian motion. We first define
weak convergence of continuous stochastic processes, and we define and prove the tools
needed to show weak convergence. Then we use the representation of fractional Brownian
motion due to Mandelbrot and Van Ness as an inspiration for a discrete stochastic process.
We use linear interpolation to extend this process to a continuous process. As the time-
intervals in the approximation becomes smaller, our processes converge weakly to the …