scholarly journals Approximating Distribution Functions by Iterated Function Systems

2002 ◽  
Author(s):  
Stefano Maria Iacus ◽  
Davide La Torre
Keyword(s):  
Iterated Function Systems ◽  
Distribution Functions ◽  
Iterated Function

scholarly journals Approximating distribution functions by iterated function systems

2005 ◽  
Vol 2005 (1) ◽  
pp. 33-46 ◽  
Author(s):  
Stefano Maria Iacus ◽  
Davide La Torre
Keyword(s):  
Monte Carlo ◽  
Distribution Function ◽  
Iterated Function System ◽  
Asymptotic Properties ◽  
Iterated Function Systems ◽  
Monte Carlo Analysis ◽  
Distribution Functions ◽  
Density Estimator ◽  
New Class ◽  
Iterated Function

An iterated function system (IFS) on the space of distribution functions is built with the aim of proposing a new class of distribution function estimators. One IFS estimator and its asymptotic properties are studied in detail. We also propose a density estimator derived from the IFS distribution function estimator by using Fourier analysis. Relative efficiencies of both estimators, for small and moderate sample sizes, are presented via Monte Carlo analysis.

scholarly journals Sampling and interpolation of cumulative distribution functions of Cantor sets in [0, 1]

2021 ◽  
Vol 54 (1) ◽  
pp. 85-109
Author(s):  
Allison Byars ◽  
Evan Camrud ◽  
Steven N. Harding ◽  
Sarah McCarty ◽  
Keith Sullivan ◽  
...  
Keyword(s):  
Cumulative Distribution Function ◽  
Invariant Measures ◽  
Borel Probability Measure ◽  
Iterated Function Systems ◽  
Distribution Functions ◽  
Cantor Sets ◽  
Cumulative Distribution ◽  
Sampling Schemes ◽  
Iterated Function ◽  
Borel Probability

Abstract Cantor sets are constructed from iteratively removing sections of intervals. This process yields a cumulative distribution function (CDF), constructed from the invariant Borel probability measure associated with their iterated function systems. Under appropriate assumptions, we identify sampling schemes of such CDFs, meaning that the underlying Cantor set can be reconstructed from sufficiently many samples of its CDF. To this end, we prove that two Cantor sets have almost-nowhere intersection with respect to their corresponding invariant measures.

scholarly journals Hölder-differentiability of Gibbs distribution functions

2009 ◽  
Vol 147 (2) ◽  
pp. 489-503 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
BERND O. STRATMANN
Keyword(s):  
Distribution Function ◽  
Gibbs Measures ◽  
Thermodynamic Formalism ◽  
Iterated Function Systems ◽  
Gibbs Distribution ◽  
Distribution Functions ◽  
Limit Sets ◽  
Iterated Function ◽  
Natural Home ◽  
Set Of Points

AbstractIn this paper we give non-trivial applications of the thermodynamic formalism to the theory of distribution functions of Gibbs measures (devil's staircases) supported on limit sets of finitely generated conformal iterated function systems in ℝ. For a large class of these Gibbs states we determine the Hausdorff dimension of the set of points at which the distribution function of these measures is not α-Hölder-differentiable. The obtained results give significant extensions of recent work by Darst, Dekking, Falconer, Li, Morris and Xiao. In particular, our results clearly show that the results of these authors have their natural home within the thermodynamic formalism.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

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