Introduction

Abstract

This chapter is an introduction for the methods and content of the book. After a brief description of the book’s contents, we give results in a simple setting: the underlying process X is a one-dimensional Lévy process which is either continuous, or has finitely many jumps in finite intervals. The process is discretized along a regular grid of mesh Δ _n which eventually goes to 0, and we introduce two kinds of functionals of interest for this setting:

• 1. The “unnormalized functional” $V^{n}(f,X)_{t}=\sum_{i=1}^{[t/\varDelta _{n}]}f(X_{i\varDelta _{n}}-X_{(i-1)\varDelta _{n}})$ for any given test function f, and where [t/Δ _n ] denotes the integer part of t/Δ _n .

• 2. The “normalized functional” $V'^{n}(f,X)_{t}=\varDelta _{n}\sum_{i=1}^{[t/\varDelta _{n}]}f((X_{i\varDelta _{n}}-X_{(i-1)\varDelta _{n}})/\sqrt{\varDelta _{n}}\,)$, where the (inside) normalized factor $\sqrt{\varDelta _{n}}$ is chosen so that when X is a Brownian motion the argument of f in each summand is a standard normal variable.

Then, with X as described above, we explain the sort of limiting behavior one may expect for these functionals: the convergence in probability towards a suitable limit, and the associated Central Limit Theorem. The simple setting allows one to give a heuristic explanation of the results, and of the conditions on the test function f which are necessary to obtain these results.