Intro
Preface
Acknowledgments
Contents
1 Setting the Stage
1.1 Review of Some Abelian Results
1.2 Illustrative Examples on Nilpotent Matrix Groups
Notation
2 Introduction
2.1 Basic Question
2.2 Description of the Basic Ingredients and Results
2.3 Detailed Description of Some Special Cases
2.3.1 Word Length Radial Stable Walks
2.3.2 Walks Taking Stable-Like Steps Along One-Parameter Subgroups
2.3.3 Walks Associated with Measure in SM()
2.4 Symmetric Continuous Convolution Semigroup of Probability Measures and Lévy Processes
2.5 Prior Results

2.5.1 Functional Type Limit Theorems
2.5.2 Local Limit Theorems
3 Polynomial Coordinates and Approximate Dilations
3.1 Polynomial Coordinate Systems
3.2 Dilations, Approximate Dilations, and Limit Groups
3.2.1 Straight Dilations
3.2.2 Approximate Group Dilations and Limit Groups
4 Vague Convergence and Change of Group Law
4.1 Vague Convergence Under Rescaling
4.2 Vague Convergence of Jump Measures and Kernels
5 Weak Convergence of the Processes
5.1 Assumption (A)
5.2 Geometries on Rd and G
5.3 Convergence of Volume
5.4 Further Hypotheses

5.4.1 The Random Walk on (Regularity)
5.4.2 Exit Time Estimates
5.4.3 Tails Properties for Jt and J
5.5 Weak Convergence
5.6 Proof of Theorem 5.11
6 Local Limit Theorem
6.1 Assumption (R2)
6.2 Statement and Proof of the LLT
7 Symmetric Lévy Processes on Nilpotent Groups
7.1 The Problem of Identifying the Limit Process
7.2 Symmetric Lévy Processes and Their Approximations
7.3 Examples
8 Measures in SM() and Their Geometries
8.1 Probability Measures in SM and SM1
8.2 Weight Systems on Associated with Measures in SM()