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Theorem 2 - Collage theorem

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

Let (Y, d_{Y} ) be a complete metric space. Given an

suppose that there exists a contractive map f on Y with contractivity factor 0 ≤ c < 1 such that

If is the fixed point of

then

So if one wishes to approximate a function y with the fixed point of an unknown contractive map f, it is only needed to solve the inverse problem of finding f which minimizes the collage distance d_{Y} (y,f(y)). The main result in Forte and Vrscay that we will use to build one of the IFSs estimators is that the inverse problem can be reduced to minimize a suitable quadratic form in terms of the p_{i} given a set of affine maps w_{i} and the sequence of moments g_{k} of the target measure. Let

be the simplex of probabilities. Let w = (w_{1},w_{2}, . . . ,w_{n}), N = 1, 2, . . . be subsets of W = {w_{1},w_{2}, . . .} the infinite set of affine contractive maps on X = [0, 1] and let g the set of the moments of any order of

Denote by M the Markov operator of the N-maps IFS (w, p) and by

, with associated moment vector of any order h_{N}. The collage distance between the moment vector of

is a continuous function and attains an absolute minimum value Δ min on Π^{N}

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