It is well known that the velocity dispersion of nearby stars increases systematically with age, and this fact is conventionally explained by postulating that stars are randomly accelerated by encounters with massive gas clouds (Spitzer & Schwarzschild 1951, 1953; Wielen 1977). However, a strong constraint on the possible role of random accelerations is provided by the apparent existence of fairly old moving groups of stars (Eggen 1969, Boyle & McClure 1975). An alternative explanation (Tinsley & Larson 1978) is that the age dependence of the velocity dispersion of stars older than 109yr is produced by a gradual decay with time of interstellar turbulent motions, as predicted by plausible collapse models. This effect cannot account directly for the variation of velocity dispersion with age observed for stars younger than 109yr, but Tinsley & Larson suggested that this could be explained if the velocity dispersion of the youngest stars reflects only the local turbulent motions of the gas, while the velocity dispersion of older stars reflects in addition larger-scale non-circular motions in the galactic gas layer. If the interstellar medium possesses a hierarchy of motions whose velocity dispersion increases with the size of the region considered, older stars which have traveled farther since their formation will sample the gas motions over a larger volume of space, and thus will have a larger velocity dispersion than the younger stars.