This paper examines properties of a class of complex-valued stable processes which have spectral representation by means of independent-increments processes. A representation is derived by an application of Schilder's stochastic integral. Also, another construction of harmonizable stable processes by means of generalized stochastic processes is given, and its relation to the stochastic integral is shown. Some limit theorems of the Fourier transform of a sample from harmonizable stable processes are provided. Moreover, a linear prediction theory which pertains to those processes is suggested as an extension of that of second-order stationary processes.