Subordination methods for free deconvolution

Arizmendi, Octavio; Tarrago, Pierre; Vargas, Carlos

doi:10.1214/20-aihp1050

Cited by 8 publications

(24 citation statements)

References 45 publications

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“…In this section, we summarize some examples of FQID distributions. Section 4.1: Using the methods in [1] and complex analysis, we obtain the following FQID distributions.…”

Section: Discussionmentioning

confidence: 99%

“…We can rephrase this in terms of the characteristic triplets. We seek a characteristic triplet for the QID distribution µ in terms of the Lévy-Khintchine representation in the sense of (1). Here, we have to be a little careful.…”

Section: Quasi-infinitely Divisible Distributionsmentioning

confidence: 99%

“…holds for z in some domain of C + . The operation ⊟ is called free deconvolution (see [1] for more information).…”

Section: Definition and Convolution Propertiesmentioning

confidence: 99%

“…We investigate whether there exist FQID probability measures whose quasi-free Lévy measure is a signed measure. This idea comes from the results in [1]. If µ ∈ P(R) has a finite variance σ 2 , then by [1, Theorem 1.2], there exists some ρ(µ) ∈ P(R) such that…”

Section: Free Deconvolution With the Cauchy Distributionmentioning

confidence: 99%

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On freely quasi-infinitely divisible distributions

Hotta¹,

Młotkowski²,

Sakuma³

et al. 2021

Preprint

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Inspired by classical quasi-infinitely divisible distributions, this paper introduces freely quasi-infinitely divisible (FQID) distributions on R. The FQID distributions are characterized by the free Lévy-Khintchine-type representation with a signed Lévy measure. Based on the representation form, we obtain various examples and distributional properties of FQID distributions. Moreover, interesting facts about FQID distributions that do not hold in the classical framework are observed-some FQID distributions admit a negative Gaussian part, and the total mass of the signed Lévy measure for some FQID distributions may be negative.

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“…In this section, we summarize some examples of FQID distributions. Section 4.1: Using the methods in [1] and complex analysis, we obtain the following FQID distributions.…”

Section: Discussionmentioning

confidence: 99%

Section: Quasi-infinitely Divisible Distributionsmentioning

confidence: 99%

“…holds for z in some domain of C + . The operation ⊟ is called free deconvolution (see [1] for more information).…”

Section: Definition and Convolution Propertiesmentioning

confidence: 99%

Section: Free Deconvolution With the Cauchy Distributionmentioning

confidence: 99%

See 2 more Smart Citations

On freely quasi-infinitely divisible distributions

Hotta¹,

Młotkowski²,

Sakuma³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…for example µ = W(−1) MP(1, 8) means that µ W(1) = MP (1,8). Note that µ is freely infinitely divisible if and only if 0 ≤ x ≤ 1.…”

Section: Wigner Law and Marchenko-pastur Lawmentioning

confidence: 99%

Probability distributions with rational free $R$-transform

Młotkowski¹

2021

Preprint

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We study the class M ratio of those probability distributions for which the free R-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the monotone convolution. We prove a sufficient condition that a rational function is the free R-transform of a probability distribution. Several examples are provided, including that of free deconvolution.

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Spectral deconvolution of matrix models: the additive case

Tarrago

2023

Information and Inference: A Journal of the IMA

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We implement a complex analytic method to build an estimator of the spectrum of a matrix perturbed by the addition of a random matrix noise in the free probabilistic regime. This method, which has been previously introduced by Arizmendi, Tarrago and Vargas, involves two steps: the first step consists in a fixed point method to compute the Stieltjes transform of the desired distribution in a certain domain, and the second step is a classical deconvolution by a Cauchy distribution, whose parameter depends on the intensity of the noise. This method thus reduces the spectral deconvolution problem to a classical one. We provide explicit bounds for the mean squared error of the first step under the assumption that the distribution of the noise is unitary invariant. In the case where the unknown measure is sparse or close to a distribution with a density with enough smoothness, we prove that the resulting estimator converges to the measure in the $1$-Wasserstein distance at speed $O(1/\sqrt{N})$, where $N$ is the dimension of the matrix.

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Subordination methods for free deconvolution

Cited by 8 publications

References 45 publications

On freely quasi-infinitely divisible distributions

On freely quasi-infinitely divisible distributions

Probability distributions with rational free $R$-transform

Spectral deconvolution of matrix models: the additive case

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