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The uniform laws of large numbers for the chaotic logistic map

Jongsig Bae1 · Changha Hwang2 · Doobae Jun3

1Department of Mathematics, Sungkyunkwan University
2Department of Applied Statistics, Dankook University
3Department of Mathematics and RINS, Gyeongsang National University
Correspondence to: Jongsig Bae
Professor, Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea. Correspondence: jsbae@skku.edu
Received October 24, 2017; Revised November 9, 2017; Accepted November 13, 2017.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The standard logistic map is an iterative function, which forms a discrete-time dynamic system. The chaotic logistic map is a kind of ergodic map defined over the unit interval. In this paper we study the limiting behaviors on the several processes induced by the chaotic logistic map. We derive the law of large numbers for the process induced by the chaotic logistic map. We also derive the uniform law of large numbers for this process. When deriving the uniform law of large numbers, we study the role of bracketing of the indexed class of functions associated with the process. Then we apply the idea of DeHardt (1971) associated with the bracketing method to the process induced by the logistic map. We finally illustrate an application to Monte Carlo integration.
Keywords : Chaotic logistic map, law of large numbers, logistic map, uniform law of large numbers

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September 2018, 29 (5)