East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
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OPTIMAL APPROXIMATION BY ONE GAUSSIAN FUNCTION TO PROBABILITY DENSITY FUNCTIONS

  • Gwang Il Kim (Department of Mathematics, Gyeongsang National University) ;
  • Seung Yeon Cho (Department of Mathematics, Gyeongsang National University) ;
  • Doobae Jun (Department of Mathematics, Gyeongsang National University)
  • Received : 2023.08.04
  • Accepted : 2023.09.30
  • Published : 2023.09.30
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Abstract

In this paper, we introduce the optimal approximation by a Gaussian function for a probability density function. We show that the approximation can be obtained by solving a non-linear system of parameters of Gaussian function. Then, to understand the non-normality of the empirical distributions observed in financial markets, we consider the nearly Gaussian function that consists of an optimally approximated Gaussian function and a small periodically oscillating density function. We show that, depending on the parameters of the oscillation, the nearly Gaussian functions can have fairly thick heavy tails.

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Acknowledgement

This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (No. NRF-2021R1I1A3048350, RS-2022-00166144, NRF-2021R1A2C1014001).

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