Asymptotic properties of the fixed points of both
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 TN as well. But if we let N varying with the sample size n we can have much more, at least from
The fixed point
of the above operator,
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 Nn 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
Next: Theorem 10.