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Theorem 7

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

Given a set of N maps and probabilities (**w**, **p**), satisfying the properties i)- v) along with W_{i} : [0, 1] →[C_{i}, D _{i}) then the fixed points of M : M([0, 1]) → M([0, 1]) and *T _{p}* : F([0, 1]) → F([0, 1]), say ~µ and ~ F respectively, relate as follows

Proof. From the contractivity of M and *T _{p}*, there exist

and

fixed points of M and *T _{p}*, respectively. Let

the thesis consists of proving

So we have:

By the uniqueness of the fixed points, we get

The previous theorem allows to reuse the results of Forte and Vrscay (1995) and in particular gives another way of finding the solution of ( **P **) in terms of ( **Q** ) at least on the simplex Π* ^{N}* by letting δ

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