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The Hadamard product and the free convolutions

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  • Chakrabarty, Arijit

Abstract

It is shown that if a probability measure ν is supported on a closed subset of (0,∞), that is, its support is bounded away from zero, then the free multiplicative convolution of ν and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained.

Suggested Citation

  • Chakrabarty, Arijit, 2017. "The Hadamard product and the free convolutions," Statistics & Probability Letters, Elsevier, vol. 127(C), pages 150-157.
    • Handle: RePEc:eee:stapro:v:127:y:2017:i:c:p:150-157
    DOI: 10.1016/j.spl.2017.04.009
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    Cited by:

    1. Huang, Zhiwen & Li, Tong & Huang, Kexin & Ke, Hanbing & Lin, Mei & Wang, Qiuwang, 2022. "Predictions of flow and temperature fields in a T-junction based on dynamic mode decomposition and deep learning," Energy, Elsevier, vol. 261(PA).

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