Limit processes for the sequences of stochastic processes defined by co-spectral and quadrature spectral distribution functions are found using the theory of weak convergence. The limit processes are shown to be Gaussian with independent increments and with covariance functions defined in terms of hypothesized spectral densities. Section 3 contains a discussion of the moments of the processes. The first and second asymptotic moments, which characterize the limit processes, are computed giving results analogous to those of Grenander and Rosenblatt [6] and Ibragimov [8] for autospectra. We also evaluate the higher asymptotic moments and put bounds on the higher moments. The latter are required in demonstrating tightness of the measures generated in $C\lbrack 0, \pi\rbrack$ by the co-spectral and quadrature spectral distribution functions. In Section 4, limit processes, under certain conditions listed in Section 2, are obtained and described in Theorem 4.5. Finally, Section 5 contains a discussion of asymptotic goodness-of-fit testing for spectral distribution functions.