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4.1 Asymptotic results for the quantile-based IFS estimator

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

Asymptotic properties of the fixed points of both

and

derive as a natural consequence, by the properties of the empirical moments and quantiles. So, one can expect that, for a fixed number of *N* maps, the fixed point of

is a consistent estimator of the fixed point of M as the sample size increases and that the fixed point of

converges to the fixed point of *T _{N}* as well. But if we let

The fixed point

of the above operator,

satisfies

for real x. The following (Glivenko-Cantelli) theorem states that

has the same properties of an admissible perturbation of the e.d.f in the sense of Winter (see Winter 1973, 1979 and Yukish, 1989). Let us denote by *N _{n}* the number of maps and coefficients in the IFS so to put in evidence the dependency of the sample size 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