Wikipedia
Existence
The population Pearson correlation coefficient is defined in terms of moments, and therefore exists for any bivariate probability distribution for which the population covariance is defined and the marginal population variances are defined and are non-zero. Some probability distributions such as the Cauchy distribution have undefined variance and hence ρ is not defined if X or Y follows such a distribution. In some practical applications, such as those involving data suspected to follow a heavy-tailed distribution, this is an important consideration. However, the existence of the correlation coefficient is usually not a concern; for instance, if the range of the distribution is bounded, ρ is always defined.
翻译
皮尔逊相关系数是根据矩定义的，因此，总体协方差及边际总体方差存在且非零的任何二元概率分布，都存在皮尔逊相关系数。某些概率分布（如柯西分布）方差不存在，因此，如果X或Y遵循此类分布，则ρ不存在。（后边的翻译尚未认真校对）在一些实际应用中，例如那些涉及怀疑遵循重尾分布的数据的应用中，这是一个重要的考虑因素。然而，相关系数的存在通常不是一个问题；例如，如果分布范围有界，则始终定义ρ。