Second Extension: Functions of Several Increments

The aim of this chapter is to extend the Laws of Large Numbers to functionals in which each summand depends on several successive increments of the underlying process

X

. This covers two different situations:

1.

The test function

f

is replaced by a function

F

on (ℝ

d

)

k

, where

d

is the dimension of

X

and

k

≥2 is an integer. Then the

i

th summand in the unnormalized functional is

$F(X_{i \varDelta _{n}}-X_{(i-1) \varDelta _{n}},\dots,X_{(i+k-1)\varDelta _{n}},-X_{(i-k-2) \varDelta _{n}})$

, and the same for the normalized functional, upon dividing each increment by

$\sqrt {\varDelta _{n}}$

.

2.

Each summand is a function of

k

n

successive increments (after dividing by

$\sqrt {\varDelta _{n}}$

for the normalized functionals), where

k

n

is a sequence of integers increasing to ∞, but such that

k

n

Δ

n

→0. This poses a formulation problem which is presented in Sect. 8.1, because then the test function must depend on

n

because its argument is

k

n

successive increments, and a form of “compatibility” for different values of

n

has to be assumed.

In Sects. 8.2 and 8.3 the Laws of Large Numbers for the unnormalized functionals are presented, for a fixed number

k

or an increasing number

k

n

of increments, respectively: the methods and results are deeply different in the two cases. In contrast, the results for the normalized functionals, given in Sect. 8.4, are basically the same for a fixed number

k

or an increasing number

k

n

of increments.

Of particular interest is the case of a fixed number

k

of increments, when the test function has a product form, for example when

d

=1 it could be

$f(x_{1},\dots,x_{k})=|x_{1}|^{p_{1}}\cdots|x_{k}|^{p_{k}}$

for positive reals

p

j

. The associated functionals are then called

multipower variations

and have been extensively used for estimating the integrated volatility when the process

X

has jumps. This application is presented from the consistency viewpoint here. The Central Limit Theorem is studied later and is given in Sect. 8.5.