Conditioned Random Walk in Weyl Chambers and Renewal Theory [chapter]

C. Lecouvey, E. Lesigne, M. Peigné
2013 Springer Proceedings in Mathematics & Statistics  
We present here the main result from [7] and explain how to use Kashiwara crystal basis theory to associate a random walk to each minuscule irreducible representation V of a simple Lie algebra; the generalized Pitman transform defined in [1] for similar random walks with uniform distributions yields yet a Markov chain when the crystal attached to V is endowed with a probability distribution compatible with its weight graduation. The main probabilistic argument in our proof is a quotient version
more » ... of a renewal theorem that we state in the context of general random walks in a lattice [7] . We present some explicit examples, which can be computed using insertion schemes on tableaux described in [8]. Introduction The Pitman transform for the Brownian motion Let (B(t)) t≥0 be a standard Brownian motion on R starting at 0. We denote by m(t) the minimum process defined by m(t) := inf 0≤s≤t B(s). The Pitman transform of (B(t)) t≥0 is given by The reader will find a proof of the following statement in [11]:
doi:10.1007/978-3-642-38806-4_11 fatcat:oiv46prgwrcdfbrajczshkvcjy