A square bias transformation: properties and applications

The properties of the square bias transformation are studied, in particular, the precise moment-type estimate for the $L_1$-metric between the transformed and the original distributions is proved, a relation between their characteristic functions is found. As a corollary, some new moment-type estimates for the proximity of arbitrary characteristic function with zero mean and finite third moment to the normal one with zero mean and the same variance are proved involving the double integrals of the square- and zero- bias transformations.