KULLBACK-LEIBLER INFORMATION MEASURE FOR STUDYING CONVERGENCE-RATES OF DENSITIES AND DISTRIBUTIONS

The objective is to propose the Kullback-Leibler (KL) information meas ure I(f1 : f2) as an index for studying rates of convergence of densit ies and distribution functions. To this end, upper bounds in terms of I(f1 : f2) for several distance functions for densities and for distri bution functions are obtained first. Many illustrations of the use of this technique are given. It is shown, for example, that the sequence of KL information measures converges to zero more slowly for a normali zed sequence of gamma random variables converging to its limiting norm al distribution than for a normalized sequence of largest order statis tics from an exponential distribution converging to its limiting extre me value distribution. Furthermore, a sequence of KL information measu res for log-normal random variables approaching normality converges mo re slowly to zero than for a sequence of normalized gamma random varia bles.