The size of singular component and shift inequalities (Q1807198)

The question of how large can be the size of singular component of translates \(\mu^{\infty}_h\) of product probability \(\mu^{\infty}\) is investigated in terms of Fisher information. Shift inequalities of the form \[ S_h(\mu^{\infty}(A))\leq \mu^{\infty}_h(A)\leq R_h(\mu^{\infty}(A)) \] are obtained for suitable functions \(R_h\) and \(S_h.\)