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Theorem 1

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

Set X = [0, 1] and let µ and

with associated moments of any order gk and

Then, the following statements are equivalent

(here C(X) is the space of continuous functions on X). This theorem gives a way to find and appropriate set of maps and probabilities by solving the so called problem of moment matching.

With the solution in hands, given the convergence of the moments, we also have the convergence of the measures and then the stationary measure of M approximates with given precision (in a sense specified by the collage theorem below) the target measure µ (see Barnsley and Demko, 1985). Next theorem, called the collage theorem is a standard result of IFS theory and is a consequence of 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