Strong approximation of fractional Brownian motion by moving averages of simple random walks

Abstract

The fractional Brownian motion is a generalization of ordinary Brownian motion, used particularly when long-range dependence is required. Its explicit introduction is due to B.B. Mandelbrot and J.W. van Ness (1968) as a self-similar Gaussian process $\WH (t)$ with stationary increments. Here self-similarity means that $(a^{-H}\WH(at): t \ge 0) \stackrel{d}{=} (\WH(t): t \ge 0)$, where $H\in (0, 1)$ is the Hurst parameter of fractional Brownian motion. F.B. Knight gave a construction of ordinary Brownian motion as a limit of simple random walks in 1961. Later his method was simplified by P. R\'ev\'esz (1990) and then by the present author (1996). This approach is quite natural and elementary, and as such, can be extended to more general situations. Based on this, here we use moving averages of a suitable nested sequence of simple random walks that almost surely uniformly converge to fractional Brownian motion on compacts when $H \in (\quart , 1)$. The rate of convergence proved in this case is $O(N^{-\min(H-\quart,\quart)}\log N)$, where $N$ is the number of steps used for the approximation. If the more accurate (but also more intricate) Koml\'os, Major, Tusn\'ady (1975, 1976) approximation is used instead to embed random walks into ordinary Brownian motion, then the same type of moving averages almost surely uniformly converge to fractional Brownian motion on compacts for any $H \in (0, 1)$. Moreover, the convergence rate is conjectured to be the best possible $O(N^{-H}\log N)$, though only $O(N^{-\min(H,\half)}\log N)$ is proved here.