  We construct the conditional versions of a multidimensional random walk given
that it does not leave the Weyl chambers of type C and of type D, respectively,
in terms of a Doob h-transform. Furthermore, we prove functional limit theorems
for the rescaled random walks. This is an extension of recent work by
Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl
chamber of type A. Our proof follows recent work by Denisov and Wachtel who
used martingale properties and a strong approximation of random walks by
Brownian motion. Therefore, we are able to keep minimal moment assumptions.
Finally, we present an alternate function that is amenable to an h-transform in
the Weyl chamber of type C.
