Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
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The uniform laws of large numbers for the chaotic logistic map

  • Bae, Jongsig (Department of Mathematics, Sungkyunkwan University) ;
  • Hwang, Changha (Department of Applied Statistics, Dankook University) ;
  • Jun, Doobae (Department of Mathematics and RINS, Gyeongsang National University)
  • Received : 2017.10.24
  • Accepted : 2017.11.13
  • Published : 2017.11.30
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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.

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Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

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