  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.
