In this paper, we introduce a joint central limit theorem (CLT) for specific bilinear forms, encompassing the resolvent of the sample covariance matrix under an elliptical distribution. Through an exhaustive exploration of our theoretical findings, we unveil a phase transition in the limiting parameters that relies on the moments of the random radius in our derived CLT. Subsequently, we employ the established CLT to address two statistical challenges under elliptical distribution. The first task involves deriving the CLT for eigenvector statistics of the sample covariance matrix. The second task aims to ascertain the limiting properties of the spiked sample eigenvalues under a general spiked model. As a byproduct, we discover that the eigenmatrix of the sample covariance matrix under a light-tailed elliptical distribution satisfies the necessary conditions for asymptotic Haar, thereby extending the Haar conjecture to broader distributions.

翻译：本文针对椭圆分布下样本协方差矩阵预解式的特定双线性形式，提出了一个联合中心极限定理。通过对理论结果的深入探究，我们发现该中心极限定理中依赖随机半径矩的极限参数存在相变现象。进而，我们利用所建立的联合中心极限定理解决了椭圆分布下的两个统计难题。第一个任务是推导样本协方差矩阵特征向量统计量的中心极限定理，第二个任务是明确一般尖峰模型中样本尖峰特征值的极限性质。作为副产品，我们发现轻尾椭圆分布下样本协方差矩阵的特征向量矩阵满足渐近Haar条件的充分性，从而将Haar猜想推广至更广泛的分布情形。