A random process on the circle is a family of random variables X(P,t) indexed by the position P on the unit circle and by the time t (t = 0, + 1, ···). For a homogeneous and stationary process, using its Fourier series expansion, we deduce a spectral representation of the covariance function. The purpose of the paper is to develop a spectral analysis when X(P,t) is observed at all the points on the circle at t = 0, 1, ···, T – 1. The asymptotic distribution of the family of finite Fourier transforms, of the family of periodograms and of the family of spectral densities estimates are obtained from results available for vector-valued time series. Also, a sample covariance function for X (P,t) is defined and its asymptotic distribution derived.
