  We consider the diffusion scaling limit of the vicious walker model that is a
system of nonintersecting random walks. We prove a functional central limit
theorem for the model and derive two types of nonintersecting Brownian motions,
in which the nonintersecting condition is imposed in a finite time interval
$(0,T]$ for the first type and in an infinite time interval $(0,\infty)$ for
the second type, respectively. The limit process of the first type is a
temporally inhomogeneous diffusion, and that of the second type is a temporally
homogeneous diffusion that is identified with a Dyson's model of Brownian
motions studied in the random matrix theory. We show that these two types of
processes are related to each other by a multi-dimensional generalization of
Imhof's relation, whose original form relates the Brownian meander and the
three-dimensional Bessel process. We also study the vicious walkers with wall
restriction and prove a functional central limit theorem in the diffusion
scaling limit.
