Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Limiting distributions for eigenvalues of sample correlation matrices
  from heavy-tailed populations   Consider a $p$-dimensional population ${\mathbf x} \in\mathbb{R}^p$ with iid
coordinates in the domain of attraction of a stable distribution with index
$\alpha\in (0,2)$. Since the variance of ${\mathbf x}$ is infinite, the sample
covariance matrix ${\mathbf S}_n=n^{-1}\sum_{i=1}^n {{\mathbf x}_i}{\mathbf
x}'_i$ based on a sample ${\mathbf x}_1,\ldots,{\mathbf x}_n$ from the
population is not well behaved and it is of interest to use instead the sample
correlation matrix ${\mathbf R}_n= \{\operatorname{diag}({\mathbf
S}_n)\}^{-1/2}\, {\mathbf S}_n \{\operatorname{diag}({\mathbf S}_n)\}^{-1/2}$.
This paper finds the limiting distributions of the eigenvalues of ${\mathbf
R}_n$ when both the dimension $p$ and the sample size $n$ grow to infinity such
that $p/n\to \gamma \in (0,\infty)$. The family of limiting distributions
$\{H_{\alpha,\gamma}\}$ is new and depends on the two parameters $\alpha$ and
$\gamma$. The moments of $H_{\alpha,\gamma}$ are fully identified as sum of two
contributions: the first from the classical Mar\v{c}enko-Pastur law and a
second due to heavy tails. Moreover, the family $\{H_{\alpha,\gamma}\}$ has
continuous extensions at the boundaries $\alpha=2$ and $\alpha=0$ leading to
the Mar\v{c}enko-Pastur law and a modified Poisson distribution, respectively.
  Our proofs use the method of moments, the path-shortening algorithm developed
in [18] and some novel graph counting combinatorics. As a consequence, the
moments of $H_{\alpha,\gamma}$ are expressed in terms of combinatorial objects
such as Stirling numbers of the second kind. A simulation study on these
limiting distributions $H_{\alpha,\gamma}$ is also provided for comparison with
the Mar\v{c}enko-Pastur law.

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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: physics
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg