  We construct the noncommutative Poisson boundaries of tracial von Neumann
algebras through the ultraproducts of von Neumann algebras. As an application
of this result, we complete the proof of Kaimanovich-Vershik's fundamental
theorems regarding noncommutative entropy. We also prove the
Amenability-Trivial Boundary equivalence and Choquet-Deny-Type I equivalence
for tracial von Neumann algebras.
