Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

Zitelli, G. L.

doi:10.1007/s10959-021-01153-x

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…See, p.766 (Characterization 1) in (Bose et al, 2002). Sample Lévy Covariance ensemble in (Zitelli, 2022) serves as a specific example of such matrices.…”

Section: Heavy-tailed Entries Suppose {Xmentioning

confidence: 99%

“…They found that a class of partitions, the special symmetric partitions, play a crucial role in the moments of the LSD. Matrices whose entries satisfy conditions like (1.1), are referred to as matrices with exploding moments and have been considered by several authors ((Benaych-Georges and Cabanal-Duvillard, 2012), (Male, 2017), (Noiry, 2018), (Zitelli, 2022)). In particular, the S matrix with exploding moments have been studied in Theorem 3.2 of (Benaych-Georges and Cabanal-Duvillard, 2012), Proposition 3.1 of (Noiry, 2018), and Theorem 1 of (Zitelli, 2022).…”

mentioning

confidence: 99%

“…Matrices whose entries satisfy conditions like (1.1), are referred to as matrices with exploding moments and have been considered by several authors ((Benaych-Georges and Cabanal-Duvillard, 2012), (Male, 2017), (Noiry, 2018), (Zitelli, 2022)). In particular, the S matrix with exploding moments have been studied in Theorem 3.2 of (Benaych-Georges and Cabanal-Duvillard, 2012), Proposition 3.1 of (Noiry, 2018), and Theorem 1 of (Zitelli, 2022). Moreover, formulae for the moments of the LSD have been provided, using free probability theory and graph theory respectively.…”

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confidence: 99%

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XX^T matrices with independent entries

Bose¹,

Sen²

2023

ALEA

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Let S = XX T be the (unscaled) sample covariance matrix where X is a real p × n matrix with independent entries. It is well known that if the entries of X are independent and identically distributed (i.i.d.) with enough moments and p/n → y = 0, then the limiting spectral distribution (LSD) of 1 n S converges to a Marčenko-Pastur law. Several extensions of this result are also known. We prove a general result on the existence of the LSD of S in probability or almost surely, and in particular, many of the above results follow as special cases. At the same time several new LSD results also follow from our general result.The moments of the LSD are quite involved but can be described via a set of partitions. Unlike in the i.i.d. entries case, these partitions are not necessarily non-crossing, but are related to the special symmetric partitions which are known to appear in the LSD of (generalised) Wigner matrices with independent entries.We also investigate the existence of the LSD of S A = AA T when A is the p × n symmetric or the asymmetric version of any of the following four random matrices: reverse circulant, circulant, Toeplitz and Hankel. The LSD of 1 n S A for the above four cases have been studied in (Bose et al., 2010) when the entries are i.i.d. We show that under some general assumptions on the entries of A, the LSD of S A exists and this result generalises the existing results of (Bose et al., 2010) significantly.

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“…See, p.766 (Characterization 1) in (Bose et al, 2002). Sample Lévy Covariance ensemble in (Zitelli, 2022) serves as a specific example of such matrices.…”

Section: Heavy-tailed Entries Suppose {Xmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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XX^T matrices with independent entries

Bose¹,

Sen²

2023

ALEA

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Operator-valued Gaussian processes and their covariance kernels

Jørgensen¹,

Tian²

2023

Infin. Dimens. Anal. Quantum. Probab. Relat. Top.

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In this paper, we discuss a new framework for operator-valued Gaussian processes and their covariance kernels. Our emphasis is four-fold: (i) starting with a positive operator-valued measure (POVM) [Formula: see text], we present algorithms for constructing an associated centered, operator-valued, Gaussian process [Formula: see text] with [Formula: see text] as its covariance kernel; (ii) we present different classes of POVMs, and we examine the corresponding classes of operator-valued Gaussian processes [Formula: see text]; (iii) for the operator-valued Gaussian processes [Formula: see text] at hand, and the non-commutative framework, we present the corresponding Itô-integrals; and we (iv) outline features of the operator-valued setting which are different from the more familiar case of scalar-valued Gaussian processes.

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Traffic probability for rectangular random matrices

Zitelli

2024

Random Matrices: Theory Appl.

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This article develops a rectangular version of Male’s theory of traffic probability, in which an algebra is equipped with a trace evaluated on arbitrary graphs whose edges are labeled by elements and whose vertices are subspaces. Using the language of traffic distributions, we characterize the asymptotic behavior of independent families of rectangular random matrices which are bi-permutation invariant. In the process, we take a tour of non-commutative probabilities and their random matrix models. Special attention is paid to rectangular random matrices with independent or exchangeable entries, including the existence and description of limiting ∗-distributions for a broad range of models.

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Amalgamated Free Lévy Processes as Limits of Sample Covariance Matrices

Cited by 3 publications

References 25 publications

XX^T matrices with independent entries

XX^T matrices with independent entries

Operator-valued Gaussian processes and their covariance kernels

Traffic probability for rectangular random matrices

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