We introduce and characterize central probability distributions on Littelmann
paths. Next we establish a law of large numbers and a central limit theorem for
the generalized Pitmann transform. We then study harmonic functions on
multiplicative graphs defined from the tensor powers of finite-dimensional Lie
algebras representations. Finally, we show there exists an inverse of the
generalized Pitman transform defined almost surely on the set of infinite paths
remaining in the Weyl chamber and explain how it can be computed.Comment: 27 pages, minor corrections and a simpler definition of the Pitman
invers