## On fractal distribution function estimation and applications

# Theorem 7

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

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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:

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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_{i} = 0 in C^{N}. This is true in particular if we choose the maps as in *T*_{F} . To be more explicit: from now on the functional *T*_{p} is intended to have fixed maps **w** and all δ_{i} = 0.

Stefano M. Iacus, Davide La Torre

**Next: ** 2.3 The choice of affine maps

**Summary: **Index