A fundamental concept in multivariate statistics, sample correlation matrix, is often used to infer the correlation/dependence structure among random variables, when the population mean and covariance are unknown. A natural block extension of it, {\it sample block correlation matrix}, is proposed to take on the same role, when random variables are generalized to random sub-vectors. In this paper, we establish a spectral theory of the sample block correlation matrices and apply it to group independent test and related problem, under the high-dimensional setting. More specifically, we consider a random vector of dimension $p$, consisting of $k$ sub-vectors of dimension $p_t$'s, where $p_t$'s can vary from $1$ to order $p$. Our primary goal is to investigate the dependence of the $k$ sub-vectors. We construct a random matrix model called sample block correlation matrix based on $n$ samples for this purpose. The spectral statistics of the sample block correlation matrix include the classical Wilks' statistic and Schott's statistic as special cases. It turns out that the spectral statistics do not depend on the unknown population mean and covariance. Further, under the null hypothesis that the sub-vectors are independent, the limiting behavior of the spectral statistics can be described with the aid of the Free Probability Theory. Specifically, under three different settings of possibly $n$-dependent $k$ and $p_t$'s, we show that the empirical spectral distribution of the sample block correlation matrix converges to the free Poisson binomial distribution, free Poisson distribution (Marchenko-Pastur law) and free Gaussian distribution (semicircle law), respectively. We then further derive the CLTs for the linear spectral statistics of the block correlation matrix under general setting.

Climate change may alter the abiotic conditions associated with macrofungal occurrence, but projections for taxonomically unresolved species complexes require cautious interpretation. Here, we used an AICc-optimized MaxEnt model to characterize the climate-defined aggregate suitability envelope of records assigned to the Lyophyllum decastes (Fr.) Singer complex in Sichuan, Yunnan, and Xizang, Southwest China, under current conditions and four RCP scenarios for the 2050s and 2070s. After spatial thinning, 70 occurrence records compiled from field surveys and published sources were modeled with correlation-filtered bioclimatic variables. The optimized model showed high discrimination for the pooled occurrence dataset (mean AUC > 0.95), and mean temperature of the coldest quarter had the largest contribution (50.8%). Projected changes differed between scenarios, with increasing fragmentation and reductions in highly suitable climate space under several futures. These outputs do not represent a lineage-resolved species distribution, as the occurrence records were not uniformly verified by molecular barcodes, and the model omits host availability, vegetation, soil, land use, dispersal, and other biotic processes. We therefore interpret the maps as a hypothesis-generating, aggregate climatic envelope for the complex that can guide stratified field surveys, molecular sampling, and habitat monitoring rather than as direct evidence of taxon-specific physiological tolerances or realized distributions.

In this paper, we investigate the asymptotic behaviors of the extreme eigenvectors in a general spiked covariance matrix, where the dimension and sample size increase proportionally. We eliminate the restrictive assumption of the block diagonal structure in the population covariance matrix. Moreover, there is no requirement for the spiked eigenvalues and the 4th moment to be bounded. Specifically, we apply random matrix theory to derive the convergence and limiting distributions of certain projections of the extreme eigenvectors in a large sample covariance matrix within a generalized spiked population model. Furthermore, our techniques are robust and effective, even when spiked eigenvalues differ significantly in magnitude from nonspiked ones. Finally, we propose a powerful statistic for hypothesis testing for the eigenspaces of covariance matrices.