Characteristic functions with some powers real -- III

It is shown that, for every integer n [greater-or-equal, slanted] 2, there exist distribution functions F on the real line (which may be chosen to be of lattice type or absolutely continuous) such that (i) F has moments of all orders, and (ii) the convolutions F*r, 1 [less-than-or-equals, slant] r [less-than-or-equals, slant] n - 1, of F with itself are all asymmetric about the origin, while F*n is symmetric. This answers a question raised in Staudte and Tata (1970) and rounds off earlier work of the present author. Incidentally, new families of distribution functions with moments of all orders and with all members of the same family having the same moment sequence are obtained. (Earlier examples of such families are due to Lebesgue and Heyde.)