A Note on Distributions in the Second Chaos

Pauliina Ilmonen; Lauri Viitasaari

doi:10.3390/sym11121487

A Note on Distributions in the Second Chaos

Pauliina Ilmonen, Lauri Viitasaari

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Abstract

In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X=∑k≥1λk(ξ2k−1) , where ξk∼N(0,1) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λk≠0 infinitely often. Our work generalises some known results presented in the literature.

Original language	English
Article number	1487
Journal	SYMMETRY
Volume	11
Issue number	12
DOIs	https://doi.org/10.3390/sym11121487
Publication status	Published - 2019
MoE publication type	A1 Journal article-refereed

Keywords

Infinitely divisible distribution
Second chaos
Smooth density
Symmetrised distributions

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10.3390/sym11121487Licence: CC BY

Symmetry-11-01487Final published version, 243 KBLicence: CC BY

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abstract = "In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X=∑k≥1λk(ξ2k−1) , where ξk∼N(0,1) are independent. We compute the L{\'e}vy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λk≠0 infinitely often. Our work generalises some known results presented in the literature. ",

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AU - Viitasaari, Lauri

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N2 - In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X=∑k≥1λk(ξ2k−1) , where ξk∼N(0,1) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λk≠0 infinitely often. Our work generalises some known results presented in the literature.

AB - In this article we study basic properties of random variables X, and their associated distributions, in the second chaos, meaning that X has a representation X=∑k≥1λk(ξ2k−1) , where ξk∼N(0,1) are independent. We compute the Lévy-Khintchine representations which we then use to study the smoothness of each density function. In particular, we prove the existence of a smooth density with asymptotically vanishing derivatives whenever λk≠0 infinitely often. Our work generalises some known results presented in the literature.

KW - Infinitely divisible distribution

KW - Second chaos

KW - Smooth density

KW - Symmetrised distributions

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