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2.1 Minimization approach

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

For affine IFS there exist a simple and useful relation between the moments of probability measures on M(X). Given an N-maps IFS(w, p) with associated Markov operator M, and given a measure

then, for any continuous function

(2)

where

In our case

so we readly have a relation involving the

moments of

Let

be the moments of the two measures, with g_{0} = h_{0} = 1. Then, by (2), with f(x) = x^{k}, we have

Recursive relations for the moments and more details on polynomial IFSs can be found in Forte and Vrscay (1995). The following theorem is due to Vrscay and can be found in Forte and Vrscay (1995) as well.

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