Rates of Convergence for Functional Approximations
to Fractional Brownian Motion

Konstantopoulos, Takis (Austin, Texas, U.S.A.)
takis@ece.utexas.edu

(joint work with A. Sakhanenko and S.J. Lin)

Fractional Brownian motion has been proposed as a macroscopic model for certain types of modern high-speed communications network traffic. To better understand the ways to approximate the process (for theoretical purposes but also for purposes of simulation), we consider a stationary walk whose increments are weighted sums of i.i.d. random variables with arbitrary distribution and finite variance. We first give a criterion for the convergence of an appropriately scaled version to a fractional Brownian motion (FBM) with Hurst parameter $H \gt 1/2$. We then examine the tightness of this approximation by deriving rates of convergence that depend on the weighting coefficients. We finally apply the results to fractional ARIMA and related models.