Abstract
The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient ρ >0 and both the population dimension p and the sample size n tend to infinity with logp = o(n1/3). As 0<ρ <1, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as 0<ρ <1/2. This differs substantially from a well-known result for the independent case where ρ = 0, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of ρ where the transition occurs. If ρ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen-Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.
Original language | English (US) |
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Pages (from-to) | 3321-3374 |
Number of pages | 54 |
Journal | Annals of Probability |
Volume | 47 |
Issue number | 5 |
DOIs | |
State | Published - Sep 1 2019 |
Externally published | Yes |
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty
Keywords
- Chen-stein poisson approximation
- Gumbel distribution
- Maximum sample correlation
- Multivariate normal distribution
- Phase transition