Some Comments on Correlation Under Pathological Conditions and some Remarks About Simple Linear Regression

Abstract

It is well known that the correlation between random variables X and Y is defined only when the variances are finite, and obviously the covariance is also finite by Cauchy-Schwartz inequality. Thus under the finiteness of the variances, the correlation is defined as $\rho =\frac{Cov(X,Y)}{\sigma _X \sigma _Y}$. In this note we examine whether or not the correlation $\rho$ has any meaning if the underlying condition(s) of finiteness is(are) violated. For example, if the covariance is finite, but one or both variances are infinite (at least via some limits), then it still makes sense to speak of zero correlation. Various possibilities and the resulting consequences are outlined. An elementary approach to a simple linear regression is also given.