We study periodic Brownian paths, wrapped around the surface of a cylinder. One characteristic of such a path is its width square, w2, defined as its variance. Though the average of w2 over all possible paths is well known, its full distribution function was investigated only recently. Generalizing w2 to w(N) defined as the Nth power of the magnitude of the deviations of the path from its mean, we show that the distribution functions of these also scale and obtain the asymptotic behaviour for both large and small w(N). © 1996 IOP Publishing Ltd.