Introduction

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.