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Register a new userLimiting distributions for eigenvalues of sample correlation matrices from heavy-tailed populations
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Consider a $p$-dimensional population ${mathbf x} inmathbb{R}^p$ with iid coordinates in the domain of attraction of a stable distribution with index $alphain (0,2)$. 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 $p/nto gamma in (0,infty)$. 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 Marv{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 Marv{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 Marv{c}enko-Pastur law.
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