Combining Dimension Reduction, Distance Measures and Covariance

We consider a measure of similarity, the Bhattacharyya distance, across distributions of random variables. We develop a novel methodology based on the marriage between the Bhattacharyya distance and the Johnson Lindenstrauss Lemma, a technique for dimension reduction, providing us with a simple yet powerful tool that allows comparisons between data-sets representing any two distributions. We demonstrate a relationship between covariance and distance measures based on a generic extension of Stein's Lemma. The degree to which different markets or sub groups of securities have different measures of their corresponding distributions tells us the extent to which they are different. This can aid investors looking for diversification or looking for more of the same thing. We consider an asset pricing application and then briefly discuss how this methodology lends itself to numerous Marketstructure studies and even applications outside the realm of finance / social sciences by illustrating a biological application.