  The properties of infinite series in the Wick powers of a free field whose
two-point correlation function has a singular infrared behavior and does not
satisfy the positivity condition are investigated. If these series are defined
on an appropriate functional domain, then the fields to which they converge
satisfy all conditions of the pseudo-Wightman formalism. For series convergent
only on analytic test functions in the momentum representation, the spectral
condition is formulated using the previously introduced notion of a carrier
cone of an analytic functional. A suitable generalization of the
Paley-Wiener-Schwartz theorem is used to prove that this condition is
satisfied.
