This book develops limit theorems for a natural class of long range random walks on finitely generated torsion free nilpotent groups.  The limits in these limit theorems are Lévy processes on some simply connected nilpotent Lie groups. Both the limit Lévy process and the limit Lie group carrying this process are determined by and depend on the law of the original random walk. The book offers the first systematic study of such limit theorems involving stable-like random walks and stable limit Lévy processes in the context of (non-commutative) nilpotent groups.

Zhen-Qing Chen is a Professor of Mathematics at the University of Washington, Seattle, Washington, USATakashi Kumagai is a Professor of Mathematics  at Waseda University, Tokyo, Japan.Laurent Saloff-Coste is the Abram R. Bullis Professor of Mathematics at Cornell University, Ithaca, New York, USA.Jian Wang is a Professor of Mathematics at Fujian Normal University, Fuzhou, Fujian Province, P.R. ChinaTianyi Zheng is a Professor of Mathematics at the University of California, San Diego, California, USA

Setting the stage.- Introduction.- Polynomial coordinates and approximate dilations.- Vague convergence and change of group law.- Weak convergence of the processes.- Local limit theorem.- Symmetric Lévy processes on nilpotent groups.- Measures in SM() and their geometries.- Adapted approximate group dilations.- The main results for random walks driven by measures in SM().