On the Hellinger type distances for filtered experiments (Q1119267)

For \(p\geq 2\) we study the Hellinger type distances \[ \rho^ p_ p(P_ T,P_ T')=| (dP_ T)^{1/p}-(dP_ T')^{1/p}| \quad^ p \] in a filtered space (\(\Omega\),F,\({\mathcal F})\) where (\(\Omega\),F) is a probability space with filtration \({\mathcal F}\), P and P' are two probability measures on (\(\Omega\),F), T is a stopping time and \(R_ T\) is the restriction of the probability measure R on the \(\sigma\)-algebra \(F_ T\). We give predictable upper bounds for this distance in terms of the Hellinger process and the divergency process of order p. In particular, for the Hellinger distance we prove that \[ \rho^ 2_ 2(P_ T,P_ T')\leq 8E_ Ph_ T, \] where h is the Hellinger process and \(E_ P\) is the expectation with respect to the measure P. The proofs are based on a predictable version of Burkholder type of inequalities for martingales. We apply the results to sequences of binary experiments and to parametric families of experiments.