A joint limit theorem for the point process of the off-diagonal entries of a sample covariance matrix $\mathbf{S}$, constructed from $n$ observations of a $p$-dimensional random vector with iid components, and the Frobenius norm of $\mathbf{S}$ is proved. In particular, assuming that $p$ and $n$ tend to infinity we obtain a central limit theorem for the Frobenius norm in the case of finite fourth moment of the components and an infinite variance stable law in the case of infinite fourth moment. Extending a theorem of Kallenberg, we establish asymptotic independence of the point process and the Frobenius norm of $\mathbf{S}$. To the best of our knowledge, this is the first result about joint convergence of a point process of dependent points and their sum in the non-Gaussian case.

翻译：本文证明了基于$n$个独立同分布$p$维随机向量观测值构建的样本协方差矩阵$\mathbf{S}$的非对角元点过程与$\mathbf{S}$的Frobenius范数的联合极限定理。特别地，假设$p$和$n$趋于无穷大，我们在分量四阶矩有限时得到了Frobenius范数的中心极限定理，并在四阶矩无限时得到了无穷方差稳定律。通过推广Kallenberg的一个定理，我们建立了点过程与$\mathbf{S}$的Frobenius范数的渐近独立性。据我们所知，这是关于非高斯情形下相依点过程及其和的联合收敛的首个结果。