Abstract
Belinschi et al. [Adv. Math., 226 (2011)] proved that the normal distribution is freely infinitely divisible. This paper establishes a certain monotonicity, real analyticity and asymptotic behavior of the density of the free Lévy measure. The monotonicity property strengthens the result in Hasebe et al. [Int. Math. Res. Not. (2019)] that the normal distribution is freely selfdecomposable.
Acknowledgments
The inserted figures are drawn on Mathematica Version 12.1.1, Wolfram Research, Inc., Champaign, IL. The authors are grateful to anonymous referees for very helpful comments to improve the readability of paper.
This research is supported by JSPS Open Partnership Joint Research Projects Grant Number JPJSBP120209921. Moreover, T. Hasebe was supported by JSPS Grant-in-Aid for Young Scientists 19K14546. Y. Ueda was supported by JSPS Grant-in-Aid for Scientific Research (B) 19H01791 and JSPS Grant-in-Aid for Young Scientists 22K13925.
Citation
Takahiro Hasebe. Yuki Ueda. "On the free Lévy measure of the normal distribution." Electron. J. Probab. 28 1 - 19, 2023. https://doi.org/10.1214/23-EJP1035
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