Laws of Large Numbers: The Basic Results

In this chapter the Laws of Large Numbers for the two types of functionals

V

n

(

f

,

X

) and

V

′

n

(

X

,

f

) are provided. By this, we mean their convergence in probability, usually for the Skorokhod topology. An important feature should be mentioned: unlike the case of the “usual” Law of Large Numbers, the limit is typically not deterministic, but random.

As seen in the text, the Law of Large Numbers for the unnormalized functionals

V

n

(

f

,

X

) holds for all semimartingales, and for arbitrary discretization schemes, under appropriate conditions on the test function

f

, of course. This can be viewed as an extension of the well known property that the approximate quadratic variation of a semimartingale

X

, that is the sum of the squared increments of

X

taken along an increasing sequence of stopping times, converges in probability to the “true” quadratic variation [

X

,

X

] of

X

when the mesh of the discretization scheme goes to 0.

In contrast, the Law of Large Numbers for the normalized functionals

V

′

n

(

f

,

X

), in which the argument of the test function

f

is taken to be the increment of

X

on each discretization interval, divided by the square-root of its length, holds only for Itô semimartingales and for regular discretization schemes (although an extension is presented in Chap.

14

later, for some special irregular discretization grids). An interesting feature is that the limiting process depends only on the Brownian part of

X

, and more specifically on the volatility process.

The chapter starts with two preliminary sections, about “general” discretization schemes, and about semimartingales that have

p

-summable jumps, including an extension of Itô’s formula to functions that are not necessarily

C

2

. It also ends with a quick description of two applications, which are studied more thoroughly in the next chapters: the estimation of the “integrated volatility” for a continuous Itô semimartingale, and the detection of jumps for an Itô semimartingale.