On logarithmic derivatives of probability densities

Abstract

We construct two examples connected with the integrability of logarithmic derivatives of probability densities on the real line, in particular, with the Fisher information number.
These examples show that the Fisher information of a probability density cannot be estimated
in terms of L
1
-norms of its first and second derivatives and the maximum of the absolute value
of the second derivative. In addition, the norm of the logarithmic derivative of the density in
L
3
cannot be estimated in terms of the norms in L
1 of the derivatives of the density of any
order.