2020/03/08 by Heiny, Johannes, Yao, Jianfeng · 3 citations
#FOS: Mathematics #Primary 60B20 #Probability (math.PR) #Secondary 60F05 60F10 60G10 60G55 60G70 #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2003.03857
Consider a p-dimensional population \mathbf x ∈ℝp with iid coordinates in the domain of attraction of a stable distribution with index α∈ (0,2). Since the variance of \mathbf x is infinite, the sample covariance matrix \mathbf Sn=n-1∑i=1n \mathbf xi\mathbf x'i based on a sample \mathbf x1,…,\mathbf xn from the population is not well behaved and it is of interest to use instead the sample correlation matrix \mathbf Rn= \diag(\mathbf Sn)\-1/2 \mathbf Sn \diag(\mathbf Sn)\-1/2. This paper finds the limiting distributions of the eigenvalues of \mathbf Rn when both the dimension p and the sample size n grow to infinity such that p/n→ γ∈ (0,∞). The family of limiting distributions \Hα,γ\ is new and depends on the two parameters α and γ. The moments of Hα,γ are fully identified as sum of two contributions: the first from the classical Marčenko-Pastur law and a second due to heavy tails. Moreover, the family \Hα,γ\ has continuous extensions at the boundaries α=2 and α=0 leading to the Marčenko-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 Hα,γ are expressed in terms of combinatorial objects such as Stirling numbers of the second kind. A simulation study on these limiting distributions Hα,γ is also provided for comparison with the Marčenko-Pastur law.