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A new kind of random process, the locally stationary random process, is defined, which includes the stationary random process as a special case. Numerous examples of locally stationary random processes are exhibited. By the generalized spectral density of a random process is meant the two-dimensional Fourier transform of the covariance of the process; as is well known, in the case of stationary processes, reduces to a positive mass distribution on the line in the plane, a fact which is the gist of the familiar Wiener-Khintchine relations. In the case of locally stationary random processes, a relation is found between the covariance and the spectral density which constitutes a natural generalization of the Wiener-Khintchine relations.