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2 Theoretical background for affine IFSs

On fractal distribution function estimation and applications

Theoretical background for affine IFS

Theorem 3 (Forte and Vrscay, 1995)

Theorem 4 (Iacus and La Torre, 2001).

Theorem 5 (Iacus and La Torre, 2001)

Theorem 8 (Forte and Vrscay, 1998)

Theorem 9 (Collage Theorem for FT, (Forte and Vrscay, 1998))

4.1 Asymptotic results for the quantile-based IFS estimator

Theorem 12 (Gill and Levit, 1995)

4.2 Characteristic function and Fourier density estimation

In this section we recall some of the results from Forte and Vrscay (1995) and Iacus and La Torre (2001) concerning the IFSs setup on the the space of distribution function. LetM(X) be the set of probability measures on B(X), the -algebra of Borel subsets of X where (X, d) is a compact metric space (in our case will be X = [0, 1] and d the Euclidean metric.) In the IFS literature the following Hutchinson metric plays a crucial role

where

thus (M(X), d_{H}) is a complete metric space (see Hutchinson, 1981).

As usual, we denote by (w, p) an N-maps contractive IFS on X with probabilities or simply an N-maps IFS, that is, a set of N affine contractions maps, w = (w_{1},w_{2}, . . . ,w_{n}),

with associated probabilities p = (p_{1}, p_{1}, . . . , p_{n}),

and

The IFS has a contractivity factor defined as

Consider the following (usually called Markov) operator M :

defined as

(1)

where w_{i}^{1}

i is the inverse function of wi and ffi stands for the composition. In Hutchinson (1981) it was shown that M is a contraction mapping on (M(X), d_{H}): for all

Thus, there exists a unique measure

the invariant measure of the IFS, such that

by Banach theorem.

Stefano M. Iacus, Davide La Torre

In this paper we review some recent results concerning the approximations of distribution functions and measures on [0, 1] based on iterated function systems.

Stefano M. Iacus, Davide La Torre