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We develop a novel methodology based on the marriage between theBhattacharyya distance, a measure of similarity across distributions of randomvariables, and the Johnson-Lindenstrauss Lemma, a technique for dimensionreduction. The resulting technique is a simple yet powerful tool that allowscomparisons between data-sets representing any two distributions. The degree towhich different entities, (markets, universities, hospitals, cities, groups ofsecurities, etc.), have different distance measures of their correspondingdistributions tells us the extent to which they are different, aidingparticipants looking for diversification or looking for more of the same thing.We demonstrate a relationship between covariance and distance measures based ona generic extension of Stein's Lemma. We consider an asset pricing applicationand then briefly discuss how this methodology lends itself to numerousmarket-structure studies and even applications outside the realm of finance /social sciences by illustrating a biological application. We provide numericalillustrations using security prices, volumes and volatilities of both thesevariables from six different countries.

The Perfect Marriage and Much More: Combining Dimension Reduction, Distance Measures and Covariance