The information topology

If (Q/sub n/)/sub n/spl isin/N/ is a sequence of probability distributions, we say that (Q/sub n/)/sub n/spl isin/N/ converges to Q in information, if D(Qn/spl par/Q)/spl rarr/0 for n/spl rarr//spl infin/, where D(P/spl par/Q)=/spl int/ log(dP/dQ)dP is the information divergence from P to Q. Convergence in information has been proved in many cases of statistical importance. On the set of probability distributions M/sub +//sup 1/ (/spl Omega/,B) there exists several relevant topologies. For instance the strong topology is defined by the total variation norm /spl par/P-Q/spl par/=sup/sub |f|/spl les/1/{/spl int//sub /spl Omega//fdP-/spl int//sub /spl Omega//fdQ}. Since it has long been known that the divergence balls B(Q, r)={P/spl isin/M/sub +//sup 1/ (/spl Omega/,B) | D(P /spl par/ Q)