The sample correlation coefficient $R$ plays an important role in many statistical analyses. We study the moments of $R$ under the bivariate Gaussian model assumption, provide a novel approximation for its finite sample mean and connect it with known results for the variance. We exploit these approximations to present non-asymptotic concentration inequalities for $R$. Finally, we illustrate our results in a simulation experiment that further validates the approximations presented in this work.

翻译：样本相关系数$R$在众多统计分析中扮演着重要角色。我们在双变量高斯模型假设下研究$R$的矩，为其有限样本均值提供了一种新颖的近似，并将其与方差已知结果联系起来。利用这些近似，我们给出了$R$的非渐近集中不等式。最后，通过模拟实验验证了本文提出的近似结果，进一步说明了我们的结论。