  We develop a novel methodology based on the marriage between the
Bhattacharyya distance, a measure of similarity across distributions of random
variables, and the Johnson-Lindenstrauss Lemma, a technique for dimension
reduction. The resulting technique is a simple yet powerful tool that allows
comparisons between data-sets representing any two distributions. The degree to
which different entities, (markets, universities, hospitals, cities, groups of
securities, etc.), have different distance measures of their corresponding
distributions tells us the extent to which they are different, aiding
participants looking for diversification or looking for more of the same thing.
We demonstrate a relationship between covariance and distance measures based on
a generic extension of Stein's Lemma. We consider an asset pricing application
and then briefly discuss how this methodology lends itself to numerous
market-structure studies and even applications outside the realm of finance /
social sciences by illustrating a biological application. We provide numerical
illustrations using security prices, volumes and volatilities of both these
variables from six different countries.
